Nootan solutions for Mathematics [English] Class 9 ICSE chapter 18 - Trigonometric Ratios of Some Standard Angles and Complementary Angles [Latest edition]

Find the value of the following expression:

sin 60° cos 30° + cos 60° sin 30°

Find the value of the following expression:

cos 60° cos 45° – sin 60° sin 45°

Find the value of the following expression:

sin 45° cos 30° – cos 45° sin 30°

Find the value of the following expression:

2 sin² 30° – 3 cos² 45° + tan² 60°

Find the value of the following expression:

4 cos² 60° + 4 sin² 45° – sin² 30°

Find the value of the following expression:

7 sin 30° cos 60°

Find the value of the following expression:

10 tan 45°

Prove that:

sin 60° = 2 sin 30° cos 30°

Without using trigonometric tables find the value of 3 sin² 45° + 2 cos² 60°.

If 0 ≤ x ≤ 90°, state the numerical value of x for which sin x° = cos x°.

Find the value of the following expression:

cos 90° + cos² 45° sin 30° tan 45°

Find the value of the following expression:

cos² 45° + sin² 60° + sin² 30°

Find the value of the following expression:

sin² 30° + cos² 60°

Find the value of the following expression:

`(sin^2 45^circ + cos^2 45^circ)/(tan^2 60^circ)`

Find the value of the following expression:

`(sin 30^circ -  sin 90^circ +  2  cos 0^circ)/(tan 30^circ xx tan 60^circ)`

Without using tables verify that cos 60° = cos² 30° – sin² 30°.

Prove that `cos^2 30^circ + sin 30^circ + tan^2 45^circ = 2 1/4`.

Prove that:

4 (sin⁴ 30° + cos⁴ 60°) – 3 (cos² 45° – sin² 90°) = 2

If x = 30°, without using tables, verify that : `sin 2x = (2 tan x)/(1 + tan^2 x)`.

If 4 sin² θ – 1 = 0 and angle θ is less than 90°, find the value of θ and hence the value of cos² θ + tan² θ.

Without using mathematical tables, evaluate, `2sqrt(2)  cos 45^circ  cos 60^circ +  2sqrt(3)  sin 30^circ  tan 60^circ -  cos 0^circ`.

Without using tables find the value of

`4/3 tan^2 30^circ + sin^2 60^circ - 3 cos^2 60^circ + 3/4 tan^2 60^circ - 2 tan^2 45^circ`

If sin (A + B) = 1 and cos (A – B) = 1, find A and B.

If `sin (A - B) = cos (A + B) = 1/2`, find A and B.

If `tan(A - B) = (1)/sqrt(3)` and `tan(A + B) = sqrt(3)`, find A and B.

Use the formula, `cos θ = sqrt((1 + cos 2θ)/2`, find the value of cos 30°, it being given that cos 60° = `1/2`.

Find the value of θ from the following: 

2 sin 2θ = 1

Find the value of θ from the following:

`tan (θ - 15^circ) = 1/sqrt(3)`

Find the value of θ from the following:

cos (2θ – 50°) = sin 60°

If A = 45°, verify that sin 2A = 2 sin A cos A.

If A = 45°, verify that cos 2A = 2 cos² A – 1.

If A = 45°, verify that `sin A = sqrt((1 - cos 2A)/2)`.

If A = 60° and B = 30°, verify that sin (A – B) = sin A cos B – cos A sin B.

If A = 60° and B = 30°, verify that `tan (A - B) = (tan A - tan B)/(1 + tan A tan B)`.

Find the value of x from the following figure.

Find the value of x from the following figure.

Find the value of θ from the following figure.

Find the value of θ from the following figure.

In ΔABC, ∠B = 90°, find ∠A if BC is equal to AВ.

In ΔABC, ∠B = 90°, find ∠A if AB is `sqrt(3)` times BC.

In the following figure, CD = 20 cm, find AB and BC.

In the adjoining figure, AB = 30 cm, find AD and BC.

From the adjoining figure, find the value of θ.

From the adjoining figure, find the length of CD.

From the adjoining figure, find the length of AC.

In the adjoining figure, DE = 1.5 m, CD = 30 m, find the length of AC.

A ladder leaning against a wall makes an angle of 60° with horizontal. Its foot is 45 m away from the wall. Find the length of the ladder.

A kite is flying at a height of 30 m from the ground. The string makes 30° angle from the horizontal. Find the length of the string.

The string of a kite is 80 m long. It makes 45° angle from the horizontal. Find the height of the kite from the ground.

A vertical tree is broken by wind such that its top touches the ground at 30 m away from its foot and makes 30° angle from the ground. Find the height of tree before broken.

In the adjoining figure, CD = 48 cm, tan ф = `5/12` and tan = `3/4`. Find the length of AB and BC.

Evaluate the following:

`(sin 32^circ)/(cos 58^circ)`

Evaluate the following:

`(sec 48^circ)/("cosec"  42^circ)`

Evaluate the following:

`(tan 55^circ)/(cot 35^circ)`

Evaluate the following:

sin 10° – cos 80°

Evaluate the following:

tan 20° – cot 70°

Evaluate: 

sin 40° sin 50° – cos 40° cos 50°

Evaluate: 

tan 25° tan 65° – cot 25° cot 65°

Evaluate: 

`((sin 18^circ)/(cos 72^circ))^2 + ((sec 32^circ)/("cosec"  58^circ))^2`

Evaluate:

4 cos 80° cosec 10° + 5 sin 59° sec 31°

Evaluate:

tan 42° tan 25° tan 48° tan 65° tan 45°

Evaluate:

`(sin 40^circ)/(cos 50^circ) + (2  "cosec"  50^circ)/(sec 40^circ) - 3  cos 50^circ  "cosec"  40^circ + 1`

Evaluate:

`(tan 70^circ)/(cot 20^circ) + (sin 38^circ)/(cos 52^circ) - 4 sin^2 30^circ`

Evaluate:

`(4  cos 57^circ)/(sin 33^circ) - (sqrt(3) · cos 38^circ "cosec"  52^circ)/(tan 20^circ tan 60^circ tan 70^circ)`