# SELINA Solutions for Class 9 Maths Chapter 22 - 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(A)

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.

Given angle

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Form the following figure, find the values of :

(i) cos B

(ii) tan C

(iii) sin^{2}B
+ cos^{2}B

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

Given angle

(i)

(ii)

(iii)

(iv)

From the following figure, find the values of :

(i) cos A (ii) cosec A

(iii) tan^{2}A - sec^{2}A (iv) sin C

(v) sec C (vi) cot^{2} C -

Consider the diagram as

Given angle and

(i)

(ii)

(iii)

(iv)

(v)

(vi)

From the following figure, find the values of :

(i) sin B (ii) tan C

(iii) sec^{2} B - tan^{2}B (iv) sin^{2}C + cos^{2}C

Given angle and

(i)

(ii)

(iii)

(iv)

Given: sin A = , find :

(i) tan A(ii) cos A

Consider the diagram below:

Therefore if length of , length of

Since

Now

(i)

(ii)

From the following figure, find the values of :

(i) sin A

(ii) sec A

(iii) cos^{2} A + sin^{2}A

Given angle in the figure

Now

(i)

(ii)

(iii)

Given: cos A =

Evaluate: (i) (ii)

Consider the diagram below:

Therefore if length of , length of

Since

Now

(i)

(ii)

Given: sec A = , evaluate : sin A -

Consider the diagram below:

Therefore if length of , length of

Since

Now

Therefore

Given: tan A = , find :

Consider the diagram below:

Therefore if length of , length of

Since

Now

Therefore

Given: 4 cot A = 3 find;

(i) sin A

(ii) sec A

(iii) cosec^{2} A - cot^{2}A.

Consider the diagram below:

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

Since

(i)

(ii)

(iii)

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

Consider the diagram below:

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

Since

Now all other trigonometric ratios are

In a right-angled triangle, it is given that A is an acute angle and tan A =.

find the value of :

(i) cos A(ii) sin A(iii)

Consider the diagram below:

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

Since

(i)

(ii)

(iii)

Given: sin

Find cos + sin in terms of p and q.

Consider the diagram below:

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

Since

Now

Therefore

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

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

Consider the diagram below:

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

Since

Consider the diagram below:

Therefore if length of AC = x, length of

Since

Now

Therefore

If 5 cot = 12, find the value of : Cosec + sec

Consider the diagram below:

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

Since

Now

Therefore

If tan x = , find the value of : 4 sin^{2}x - 3 cos^{2}x + 2

Consider the diagram below:

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

Since

Now

Therefore

Ifcosec = , find the value of:

(i) 2 - sin^{2} - cos^{2}

(ii)

Consider the diagram below:

Therefore if length of hypotenuse , length of perpendicular = x

Since

Now

(i)

(ii)

If sec A = , find the value of :

Consider the diagram below:

Therefore if length of AB = x, length of

Since

Now

Therefore

If cot = 1; find the value of: 5 tan^{2} + 2 sin^{2} - 3

Consider the diagram below:

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

Since

Now

Therefore

In the following figure:

AD BC, AC = 26 CD = 10, BC = 42,

DAC = x and B = y.

Find the value of :

(i) cot x

(ii)

(iii)

Given angle and in the figure

Again

Now

(i)

(ii)

Therefore

(iii)

Therefore

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

From the following figure, find:

(i) y (ii) sin x^{o}

(iii) (sec x^{o} - tan x^{o}) (sec x^{o} + tan x^{o})

Consider the given figure

(i)

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

(ii)

(iii)

Therefore

Use the given figure to find:

(i) sin x^{o} (ii) cos y^{o}

(iii) 3 tan x^{o} - 2 sin y^{o} + 4 cos y^{o}.

Consider the given figure

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

Also

(i)

(ii)

(iii)

Therefore

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

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)

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 DBC

(ii) sin DBA

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

Now using Pythagorean Theorem

Therefore

(i)

(ii)

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

Consider the figure below

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

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) sin^{2} B + cos^{2}B (iv) tan C - cot B

Consider the figure below

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

Since

(i)

(ii)

(iii)

Therefore

(iv)

Therefore

In triangle ABC; ABC = 90^{o}, CAB = x^{o}, tan x^{o} = and BC = 15 cm. Find the measures of AB and AC.

Consider the figure

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

Since

Now

Therefore

And

Using the measurements given in the following figure:

(i) Find the value of sin and tan.

(ii) Write an expression for AD in terms of

Consider the figure

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

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

(i)

(ii)

Or

In the given figure;

BC = 15 cm and sin B =.

(i) Calculate the measure of AB and AC.

(ii) Now, if tan ADC = 1; calculate the measures of CD and AD.

Also, show that: tan^{2}B -

Given

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

Since

Now

(i)

And

(ii)

Given

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

Since

Now

So

And

Now

So

If sin A + cosec A = 2;

Find the value of sin^{2}
A + cosec^{2} A.

Squaring both sides

If tan A + cot A = 5;

Find the value of tan^{2}
A + cot^{2} A.

Squaring both sides

Given: 4 sin = 3 cos ; find the value of:

(i) sin (ii) cos

(iii) cot^{2} - cosec^{2}.

(iv) 4 cos^{2}- 3 sin^{2}+ 2

Consider the diagram below:

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

Since

(i)

(ii)

(iii)

Therefore

(iv)

Given : 17 cos = 15;

Find the value of: tan + 2 sec.

Consider the diagram below:

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

Since

Now

Therefore

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

.

Now

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

(i) (ii)

Since is mid-point of so

(i)

(ii)

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

(i) cos A(ii) 3 - cot^{2} A + cosec^{2}A.

Consider the diagram below:

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

Since

(i)

(ii)

Therefore

In triangle ABC, B = 90^{o} and tan A = 0.75. If AC = 30 cm, find the lengths of AB and BC.

Consider the figure

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

Since

Now

Therefore

And

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.

Consider the figure

The diagonals of a rhombus bisects each other perpendicularly

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

Since

Now

Therefore

And

Since the sides of a rhombus are equal so the length of the side of the rhombus

The diagonals are

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

(i) cos B (ii) sin C

(iii) tan^{2} B - sec^{2} B + 2

Consider the figure below

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

Since

(i)

(ii)

(iii)

Therefore

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.

Consider the figure below

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

Since

Now

Therefore

And

Again

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

Since

Now

Therefore

And

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

.

Now

If sin A = cos A, find the value of 2 tan^{2}A - 2 sec^{2} A + 5.

Consider the figure

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

Since

Now

Therefore

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

Consider the diagram

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

Since

Now

Therefore

Now

If 2 sin x = , evaluate.

(i) 4 sin^{3} x - 3 sin x.

(ii) 3 cos x - 4 cos^{3} x.

Consider the figure

Therefore if length of , length of

Since

Now

(i)

(ii)

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

Consider the diagram below:

Therefore if length of , length of

Since

Consider the diagram below:

Therefore if length of , length of

Since

Now

Therefore

Use the informations given in the following figure to evaluate:

Consider the given diagram as

Using Pythagorean Theorem

Now

Again using Pythagorean Theorem

Now

Therefore

If sec A = , find: .

Consider the figure

Therefore if length of , length of

Since

Now

Therefore

If 5 cos = 3, evaluate : .

Now

If
cosec A + sin A = 5, find the value of cosec^{2}A + sin^{2}A.

Squaring both sides

If 5 cos = 6 sin ; evaluate:

(i) tan (ii)

Now

(i)

(ii)

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