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

Learn with Selina Solutions for ICSE Class 9 Mathematics Chapter 22 Trigonometrical Ratios [Sine, Cosine, Tangent of an Angle and their Reciprocals]. At TopperLearning, academic experts have written these model answers to help you understand sine, cosine, cotangent, cosecant, secant and tangent in Trigonometry. You can access all of this and more in one place on our online study portal.

The Selina textbook solutions will also serve as a study support material for last-minute revision before your ICSE Class 9 Maths exam. Now, you can practise more problems related to trigonometric ratios in our Frank solutions. With enough practice, you can remember concepts effectively and score higher marks in your exam.

## 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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