SELINA Solutions for Class 9 Maths Chapter 22 - Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals]

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Chapter 22 - Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals] Exercise Ex. 22(A)

Question 1

From the following figure, find the values of :

(i) sin A

(ii) cos A

(iii) cot A

(iv) sec C

(v) cosec C

(vi) tan C.

Solution 1

Given angle

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 2

Form the following figure, find the values of :

(i) cos B

(ii) tan C

(iii) sin2B + cos2B

(iv) sin B. cos C + cos B. sin C

Solution 2

Given angle

(i)

(ii)

(iii)

(iv)

Question 3

From the following figure, find the values of :

(i) cos A (ii) cosec A

(iii) tan2A - sec2A (iv) sin C

(v) sec C (vi) cot2 C - Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 3

Consider the diagram as

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals 

Given angle Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals and Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(v)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(vi)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 4

From the following figure, find the values of :

(i) sin B (ii) tan C

(iii) sec2 B - tan2B (iv) sin2C + cos2C

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 4

Given angle Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals and Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 5

Given: sin A = Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, find :

(i) tan A(ii) cos A

Solution 5

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore if length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Now

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 6

From the following figure, find the values of :

(i) sin A

(ii) sec A

(iii) cos2 A + sin2A

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 6

Given angle Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsin the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Now

(i)Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

 

Question 7

Given: cos A = Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Evaluate: (i) Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 7

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore if length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 8

Given: sec A = , evaluate : sin A -

Solution 8

Consider the diagram below:

Therefore if length of , length of

Since

Now

Therefore

Question 9

Given: tan A = Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, find : Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 9

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 10

Given: 4 cot A = 3 find;

(i) sin A

(ii) sec A

(iii) cosec2 A - cot2A.

Solution 10

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals 

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of AB = 3x, length of BC = 4x

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 11

Given: cos A = 0.6; find all other trigonometrical ratios for angle A.

Solution 11

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of AB = 3x, length of AC = 5x

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now all other trigonometric ratios are

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 12

In a right-angled triangle, it is given that A is an acute angle and tan A =Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals.

find the value of :

(i) cos A(ii) sin A(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 12

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of AB = 12x, length of BC = 5x

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 13

Given: sin Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Find cos Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals+ sin Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsin terms of p and q.

Solution 13

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of perpendicular = px, length of hypotenuse = qx


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 14

If cos A = Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsand sin B = Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, find the value of : Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Are angles A and B from the same triangle? Explain.

Solution 14

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of AB = x, length of AC = 2x

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

 

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of AC = x, length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals 

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 15

If 5 cot Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals= 12, find the value of : Cosec Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals+ sec Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 15

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of base = 12x, length of perpendicular = 5x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 16

If tan x = Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, find the value of : 4 sin2x - 3 cos2x + 2

Solution 16

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of base = 3x, length of perpendicular = 4x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 17

Ifcosec Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals= Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, find the value of:

(i) 2 - sin2 Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals- cos2 Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 17

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals 

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of hypotenuse Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, length of perpendicular = x

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

 

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 18

If sec A = Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, find the value of :

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 18

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of AB = x, length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals 

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 19

If cot Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals= 1; find the value of: 5 tan2 Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals+ 2 sin2 Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals- 3

Solution 19

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of base = x, length of perpendicular = x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 20

In the following figure:

AD Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their ReciprocalsBC, AC = 26 CD = 10, BC = 42,

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their ReciprocalsDAC = x and Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their ReciprocalsB = y.

Find the value of :

(i) cot x

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 20

Given angle Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals and Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsin the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Again

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Chapter 22 - Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals] Exercise Ex. 22(B)

Question 1

From the following figure, find:

(i) y (ii) sin xo

(iii) (sec xo - tan xo) (sec xo + tan xo)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 1

Consider the given figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals 

(i)

Since the triangle is a right angled triangle, so using Pythagorean Theorem

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 2

Use the given figure to find:

(i) sin xo (ii) cos yo

(iii) 3 tan xo - 2 sin yo + 4 cos yo.

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 2

Consider the given figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Since the triangle is a right angled triangle, so using Pythagorean Theorem

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Also

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 3

In the diagram, given below, triangle ABC is right-angled at B and BD is perpendicular to AC. Find:

(i) cos DBC (ii) cot DBA

Solution 3

Consider the given figure

Since the triangle is a right angled triangle, so using Pythagorean Theorem

In and , the is common to both the triangles, so therefore .

Therefore and are similar triangles according to AAA Rule

So

(i)

(ii)

Question 4

In the given figure, triangle ABC is right-angled at B. D is the foot of the perpendicular from B to AC. Given that BC = 3 cm and AB = 4 cm. find:

(i) tan Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their ReciprocalsDBC

(ii) sin Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their ReciprocalsDBA

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 4

Consider the given figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals 

Since the triangle is a right angled triangle, so using Pythagorean Theorem

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

In Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsand Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, the Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsis common to both the triangles, Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsso thereforeSelina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals.

Therefore Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsand Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsare similar triangles according to AAA Rule

So

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Now using Pythagorean Theorem

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore


Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i) 

 Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

 

 

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 5

In triangle ABC, AB = AC = 15 cm and BC = 18 cm, find cos ABC.

Solution 5

Consider the figure below

In the isosceles , and the perpendicular drawn from angle to the side divides the side into two equal parts

Question 6

In the figure given below, ABC is an isosceles triangle with BC = 8 cm and AB = AC = 5 cm. Find:

(i) sin B (ii) tan C

(iii) sin2 B + cos2B (iv) tan C - cot B

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 6

Consider the figure below

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

In the isosceles Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals and Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals the perpendicular drawn from angle Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals to the side Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals divides the side Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals into two equal parts Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Since Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 7

In triangle ABC; Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their ReciprocalsABC = 90o, Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their ReciprocalsCAB = xo, tan xo = Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsand BC = 15 cm. Find the measures of AB and AC.

Solution 7

Consider the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of base = 4x, length of perpendicular = 3x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


And

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 8

Using the measurements given in the following figure:

(i) Find the value of sin Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocalsand tanSelina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals.

(ii) Write an expression for AD in terms of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 8

Consider the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

 

A perpendicular is drawn from D to the side AB at point E which makes BCDE is a rectangle.

Now in right angled triangle BCD using Pythagorean Theorem

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Since BCDE is rectangle so ED 12 cm, EB = 5 and AE = 14 - 5 = 9 

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Or

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 9

In the given figure;

BC = 15 cm and sin B =Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals.

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i) Calculate the measure of AB and AC.

(ii) Now, if tan Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their ReciprocalsADC = 1; calculate the measures of CD and AD.

Also, show that: tan2B - Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Solution 9

Given


Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of perpendicular = 4x, length of hypotenuse = 5x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

And

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Given

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of perpendicular = x, length of hypotenuse = x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

So

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

And

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


So

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 10

If sin A + cosec A = 2;

Find the value of sin2 A + cosec2 A.

Solution 10

Squaring both sides

Question 11

If tan A + cot A = 5;

Find the value of tan2 A + cot2 A.

Solution 11

Squaring both sides

Question 12

Given: 4 sin Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals= 3 cos Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals; find the value of:

(i) sin Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals(ii) cos Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii) cot2 Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals- cosec2Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals.

(iv) 4 cos2Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals- 3 sin2Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals+ 2

Solution 12

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of BC = 3x, length of AB = 4x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iv)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 13

Given : 17 cos Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals= 15;

Find the value of: tan Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals+ 2 secSelina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals.

Solution 13

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of AB = 15x, length of AC = 17x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 14

Given : 5 cos A - 12 sin A = 0; evaluate  :

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals.

Solution 14

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 15

In the given figure; C = 90o and D is mid-point of AC. Find

(i) (ii)

Solution 15

Since is mid-point of so

(i)

(ii)

Question 16

If 3 cos A = 4 sin A, find the value of :

(i) cos A(ii) 3 - cot2 A + cosec2A.

Solution 16

Consider the diagram below:

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of AB = 4x, length of BC = 3x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 17

In triangle ABC, Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their ReciprocalsB = 90o and tan A = 0.75. If AC = 30 cm, find the lengths of AB and BC.

Solution 17

Consider the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of base = 4x, length of perpendicular = 3x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


And

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 18

In rhombus ABCD, diagonals AC and BD intersect each other at point O.

If cosine of angle CAB is 0.6 and OB = 8 cm, find the lengths of the the side and the diagonals of the rhombus.

Solution 18

Consider the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

The diagonals of a rhombus bisects each other perpendicularly

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of base = 3x, length of hypotenuse = 5x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

And

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Since the sides of a rhombus are equal so the length of the side of the rhombus Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


The diagonals are

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 19

In triangle ABC, AB = AC = 15 cm and BC = 18 cm. Find:

(i) cos B (ii) sin C

(iii) tan2 B - sec2 B + 2

Solution 19

Consider the figure below

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

In the isosceles Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, the perpendicular drawn from angle Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals to the side Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals divides the side Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals into two equal parts Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Since Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 20

In triangle ABC, AD is perpendicular to BC. sin B = 0.8, BD = 9 cm and tan C = 1. Find the length of AB, AD, AC and DC.

Solution 20

Consider the figure below

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of perpendicular = 4x, length of hypotenuse = 5x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


And

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Again

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of perpendicular = x, length of base = x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


And

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 21

Given q tan A = p, find the value of :

.

Solution 21

Now

Question 22

If sin A = cos A, find the value of 2 tan2A - 2 sec2 A + 5.

Solution 22

Consider the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of perpendicular = x, length of base = x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 23

In rectangle ABCD, diagonal BD = 26 cm and cotangent of angle ABD = 1.5. Find the area and the perimeter of the rectangle ABCD.

Solution 23

Consider the diagram

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore if length of base = 3x, length of perpendicular = 2x


Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Therefore

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals


Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 24

If 2 sin x = Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, evaluate.

(i) 4 sin3 x - 3 sin x.

(ii) 3 cos x - 4 cos3 x.

Solution 24

Consider the figure

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Therefore if length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals, length of Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Since

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Now

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Trigonometrical Ratios Sine Consine Tangent Of An Angle And Their Reciprocals

Question 25

If sin A = and cos B = , find the value of : .

Solution 25

Consider the diagram below:

Therefore if length of , length of

Since

Consider the diagram below:

Therefore if length of , length of

Since

Now

Therefore

Question 26

Use the informations given in the following figure to evaluate:

Solution 26

Consider the given diagram as

Using Pythagorean Theorem

Now

Again using Pythagorean Theorem

Now

Therefore

Question 27

If sec A = , find: .

Solution 27

Consider the figure

Therefore if length of , length of

Since

Now

Therefore

Question 28

If 5 cos = 3, evaluate : .

Solution 28

Now

Question 29

If cosec A + sin A = 5, find the value of cosec2A + sin2A.

Solution 29

Squaring both sides

Question 30

If 5 cos = 6 sin ; evaluate:

(i) tan (ii)

Solution 30

Now

(i)

(ii)