Selina solutions for Concise Mathematics [English] Class 9 ICSE chapter 23 - Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] [Latest edition]

find the value of: sin 30° cos 30°

find the value of: tan 30° tan 60°

find the value of: cos² 60° + sin² 30°

find the value of: cosec² 60° - tan² 30°

find the value of: sin² 30° + cos² 30°+ cot² 45°

find the value of: cos² 60° + sec² 30° + tan² 45°

Find the value of:

tan² 30° + tan² 45° + tan² 60°

find the value of :

`( tan 45°)/ (cos ec30°) +( sec60°)/(co 45°) – (5 sin 90°)/ (2 cos 0°)`

find the value of :

3sin² 30° + 2tan² 60° - 5cos² 45°

Prove that:

sin 60° cos 30° + cos 60° . sin 30°  = 1

Prove that:

cos 30° . cos 60° - sin 30° . sin 60°  = 0

Prove that:

cosec² 45°  - cot² 45°  = 1

Prove that:

cos² 30°  - sin² 30° = cos 60°

Prove that:

`((tan60°  + 1)/(tan 60°  – 1))^2 = (1+ cos 30°) /(1– cos 30°) `

Prove that:

3 cosec² 60°  - 2 cot² 30°  + sec² 45°  = 0

prove that:

sin (2 × 30°) = `(2 tan 30°)/(1+tan^2 30°)`

prove that:

cos (2 x 30°) = `(1 – tan^2 30°)/(1+tan^2 30°)`

prove that:

tan (2 x 30°) = `(2 tan 30°)/(1– tan^2 30°)`

ABC is an isosceles right-angled triangle. Assuming of AB = BC = x, find the value of each of the following trigonometric ratios: sin 45°

ABC is an isosceles right-angled triangle. Assuming of AB = BC = x, find the value of each of the following trigonometric ratio: cos 45°

ABC is an isosceles right-angled triangle. Assuming of AB = BC = x, find the value of each of the following trigonometric ratios: tan 45°

Prove that:
sin 60° = 2 sin 30° cos 30°

Prove that:

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

If sin x = cos x and x is acute, state the value of x

If sec A = cosec A and 0° ∠A ∠90°, state the value of A

If tan θ = cot θ and 0°∠θ ∠90°, state the value of θ

If sin x = cos y; write the relation between x and y, if both the angles x and y are acute.

If sin x = cos y, then x + y = 45° ; write true of false

secθ . Cot θ= cosecθ ; write true or false

For any angle θ, state the value of: sin² θ + cos² θ

State for any acute angle θ whether sin θ increases or decreases as θ increases

State for any acute angle θ whether cos θ increases or decreases as θ increases.

State for any acute angle θ whether tan θ increases or decreases as θ decreases.

If `sqrt3` = 1.732, find (correct to two decimal place)  the value of sin 60^o

If `sqrt3` = 1.732, find (correct to two decimal place)  the value of  `(2)/(tan 30°)`

Evaluate: 

`(cos3"A" – 2cos4"A")/(sin3"A" + 2sin4"A")` , when A = 15°

Evaluate : 

`(3 sin 3"B" + 2 cos(2"B" + 5°))/(2 cos 3"B" – sin (2"B" – 10°)` ; when "B" = 20°.

Given A = 60° and B = 30°,
prove that : sin (A + B) = sin A cos B + cos A sin B

Given A = 60° and B = 30°,
prove that : cos (A + B) = cos A cos B - sin A sin B

Given A = 60° and B = 30°,
prove that : cos (A - B) = cos A cos B + sin A sin B

Given A = 60° and B = 30°,

prove that: tan (A - B) = `(tan"A"  –  tan"B")/(1 + tan"A".tan"B")`

If A =30^o, then prove that :
sin 2A = 2sin A cos A =  `(2 tan"A")/(1 + tan^2"A")`

If A =30^o, then prove that :
cos 2A = cos²A - sin²A = `(1 – tan^2"A")/(1+ tan^2"A")`

If A = 30^o, then prove that :

2 cos² A - 1 = 1 - 2 sin²A

If A =30^o, then prove that :
sin 3A = 3 sin A - 4 sin³A.

If A = B = 45° ,
show that:
sin (A - B) = sin A cos B - cos A sin B

If A = B = 45° ,
show that:
cos (A + B) = cos A cos B - sin A sin B

If A = 30°;
show that:
sin 3 A = 4 sin A sin (60° - A) sin (60° + A)

If A = 30°;
show that:
(sinA - cosA)² = 1 - sin2A

If A = 30°;
show that:
cos 2A = cos⁴ A - sin⁴ A

If A = 30°;
show that:
`(1 – cos 2"A")/(sin 2"A") = tan"A"`

If A = 30°;
show that:
`(1 + sin 2"A" + cos 2"A")/(sin "A" + cos"A") = 2 cos "A"`

If A = 30°;
show that:
4 cos A cos (60° - A). cos (60° + A) = cos 3A

If A = 30°;
show that:
`(cos^3"A" – cos 3"A")/(cos "A") + (sin^3"A" + sin3"A")/(sin"A") = 3`

Solve the following equation for A, if 2 sin A = 1

Solve the following equation for A, if 2cos2A = 1

Solve the following equation for A, if sin 3 A = `sqrt3 /2`

Solve the following equation for A, if sec 2A = 2

Solve the following equations for A, if `sqrt3` tan A = 1

Solve the following equation for A, if tan 3 A = 1

Solve the following equation for A, if 2 sin 3 A = 1

Solve the following equation for A, if `sqrt3` cot 2 A = 1

Calculate the value of A, if (sin A - 1) (2 cos A - 1) = 0

Calculate the value of A, if (tan A - 1) (cosec 3A - 1) = 0

Calculate the value of A, if (sec 2A - 1) (cosec 3A - 1) = 0

Calculate the value of A, if cos 3A. (2 sin 2A - 1) = 0

Calculate the value of A, if (cosec 2A - 2) (cot 3A - 1) = 0

If 2 sin x° - 1 = 0 and x° is an acute angle; find:

1. sin x°
2. x° 
3. cos x° and tan x°.

If 4 cos² x° - 1 = 0 and 0 ∠ x° ∠ 90°,
find:(i) x°
(ii) sin² x° + cos² x°
(iii) `(1)/(cos^2xx°) – (tan^2 xx°)`

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 + tan²θ.

If sin 3A = 1 and 0 < A < 90^°, find sin A

If sin 3A = 1 and 0 < A < 90^°, find cos 2A

If sin 3A = 1 and 0 < A < 90^°, find `tan^2A - (1)/(cos^2 "A")`

If 2 cos 2A =  `sqrt3` and A is acute,
find:
(i) A 
(ii) sin 3A
(iii) sin² (75° - A) + cos² (45° +A)

If sin x + cos y = 1 and x = 30°, find the value of y

If 3 tan A - 5 cos B = `sqrt3` and B = 90°, find the value of A

From the given figure,
find:
(i) cos x°
(ii) x°
(iii) `(1)/(tan^2 xx°) – (1)/(sin^2xx°)` 
(iv) Use tan x^o, to find the value of y.

Use the given figure to find:
(i) tan θ°
(ii)  θ°
(iii) sin²θ° - cos²θ°
(iv) Use sin θ° to find the value of x.

Find the magnitude of angle A, if 2 sin A cos A - cos A - 2 sin A + 1 = 0

Find the magnitude of angle A, if tan A - 2 cos A tan A + 2 cos A - 1 = 0

Find the magnitude of angle A, if 2 cos² A - 3 cos A + 1 = 0

Find the magnitude of angle A, if 2 tan 3A cos 3A - tan 3A + 1 = 2 cos 3A

Solve for x : 2 cos 3x - 1 = 0

Solve for x : cos  `(x)/(3)  –1` = 0

Solve for x : sin (x + 10°) = `(1)/(2)` 

Solve for x : cos (2x - 30°) = 0

Solve for x : 2 cos (3x - 15°) = 1

Solve for x : tan² (x - 5°) = 3

Solve for x : 3 tan² (2x - 20°) = 1

Solve for x : cos `(x/(2)+10°) = (sqrt3)/(2)`

Solve for x : sin² x + sin² 30° = 1

Solve for x : cos² 30° + cos² x = 1

Solve for x : cos² 30° + sin² 2x = 1

Solve for x : sin² 60° + cos² (3x- 9°) = 1

If 4 cos² x = 3 and x is an acute angle;
find the value of :
(i) x 
(ii) cos² x + cot² x
(iii) cos 3x (iv) sin 2x

In ΔABC, ∠B = 90° , AB = y units, BC = `(sqrt3)` units, AC = 2 units and angle A = x°, find:

1. sin x°
2. x°
3. tan x° 
4. use cos x° to find the value of y.

If 2 cos (A + B) = 2 sin (A - B) = 1;
find the values of A and B.