Selina solutions for Concise Mathematics [English] Class 9 ICSE chapter 21 - Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals] [Latest edition]

If sin A = `5/13` the value of tan A is ______.

If tan A = `3/5` the value of sin² A + cos² A is ______.

If cot A = `5/12` the value of cot² A × cosec² A is ______.

In the given figure (each observation is in cm) tan C is:

If 5 cos A = 3, the value of sec² A − tan² A is ______.

From the following figure, find the values of:

1. sin A
2. cos A
3. cot A
4. sec C
5. cosec C
6. tan C

Form the following figure, find the values of:

1. cos B
2. tan C
3. sin²B + cos²B
4. sin B. cos C + cos B. sin C

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 - ` 1 / sin^2 "c"`

From the following figure, find the values of
(i) sin B 
(ii) tan C
(iii) sec² B - tan²B 
(iv) sin²C + cos²C

Given : sin A = `(3)/(5)` , find : (i) tan A (ii) cos A 

From the following figure, find the values of :
(i) sin A
(ii) sec A
(iii) cos² A + sin²A

Given: cos A = `( 5 )/ ( 13 )`

Evaluate:

1. `(sin "A "–cot "A") / (2 tan "A")`
2. `cot "A" + 1/cos"A"`

Given: sec A = `( 29 )/(21), "evaluate : sin A" - 1/tan "A"`

Given: tan A = `4/3 , "find" : ("cosec""A")/(cot "A"– sec "A")`

Given: 4 cot A = 3
find :
(i) sin A
(ii) sec A
(iii) cosec²A - cot²A.

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

In a right-angled triangle, it is given that A is an acute angle and tan A = `(5) /(12)`.

find the value of :
(i) cos A
(ii) sin A
(iii)  ` (cosA+sinA)/(cosA– sin A)`

Given: sin θ = `p/q`, find cos θ + sin θ in terms of p and q.

If cos A = `(1)/(2)` and sin B = `(1)/(sqrt2)`, find the value of: `(tan"A"  –  tan"B")/(1+tan"A" tan"B")`.

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

If cos A = `1/sqrt2` and sin B = `sqrt3/2` the value of `(tan B-tan A)/(1+ tan A tan B)` is ______.

From the given figure, the value of cos y is:

ABCD is a rhombus with diagonals BD and AC equal to 12 cm and 16 cm respectively. The value of cosec x is ______.

If sin A − cosec A = 2, the value of sin² A + cosec² A is ______.

If cot A = `sqrt5`, the value of cosec² A − sec² A is ______.

From the following figure, find:
(i) y
(ii) sin x°
(iii) (sec x° - tan x°) (sec x° + tan x°)

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.

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

1. cos ∠DBC
2. cot ∠DBA

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 :

1. tan ∠DBC
2. sin ∠DBA

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

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

In triangle ABC; ∠ABC = 90°, ∠CAB = x°, tan x° = `(3)/(4)` and BC = 15 cm. Find the measures of AB and AC.

Using the measurements given in the following figure:

1. Find the value of sin Φ and tan θ.
2. Write an expression for AD in terms of θ.

In the given figure;

BC = 15 cm and sin B = `(4)/(5)`

1. Calculate the measure of AB and AC.
2. Now, if tan ∠ADC = 1; calculate the measures of CD and AD.

Also, show that: tan²B - `1/cos^2 "B" = – 1 .`

If sin A + cosec A = 2;

Find the value of sin² A + cosec² A.

If tan A + cot A = 5;
Find the value of tan² A + cot² A.

Given: 4 sin θ = 3 cos θ ; find the value of:
(i) sin θ (ii) cos θ
(iii) cot² θ - cosec² θ .
(iv) 4 cos²θ- 3 sin²θ+2

Given : 17 cos θ = 15;
Find the value of: tan θ + 2 secθ .

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

`(sin "A"+cos"A")/(2 cos"A"– sin"A")`

\[\frac{2 \tan 30° }{1 + \tan^2 30°}\]  is equal to ______.

If tan `3A - sqrt3` = 0 and 0 ≤ 3A ≤ 90°; the measure of angle A is ______.

If cot A = tan A and 0 ≤ A ≤ 90°, the measure of angle A is ______.

The value of cos² 60° − 2 sin³ 30° + 3 cot⁴ 45° is ______.

The value of `(cos 60° - cos 0° + 2sin 90°)/(cot 60° xx cot 30°)` is ______.

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:

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

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 A = 30°, then \[\frac{\sin 2A}{1 - \cos 2A}\] is equal to ______.

If A = 60° and B = 30°; the value of sin A cos B + cos A sin B is equal to ______.

If A = 30°; 3 sin A − 4 sin³ A is equal to ______.

If A = 30°, then cos⁴ A − sin⁴ A is equal to ______.

If A = 45°; then 2 sin A cos A is equal to ______.

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

If A = 30° and B = 60°, verify 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 2sin A − 1 = 0 and A is an acute angle, the measure of angle A is ______.

If cos A (cos A − 1) = 0; the measure of angle A is ______.

If tan⁴ A − 1 = 0 and angle A is acute, then A is ______.

If 2sin 3A − 1 = 0 ; the value of angle A is ______.

If 3tan² A − 1 = 0 and angle A is acute, the measure of angle A is ______.

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 + tan² θ.

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 A = cos A, the measurement of angle A is ______.

If sin A = cos B and A ≠ B then the relation between angles A and B is ______.

If A + B = 90°, the value of \[\frac{\cos A}{\sin B} \times \frac{\tan B}{\cot A}\] is ______.

The value of: cosec 40° cos 50° + sin 50° sec 40° is ______.

In a triangle ABC, sec \[\frac{A + C}{2}\] is equal to ______.

Evaluate : `(cos22°)/(sin 68°)`

Evaluate: `(tan 47°)/(cot 43°)`

Evaluate: `(sec 75°)/("cosec" 15°)`

Evaluate: `(cos55°)/(sin 35°) + (cot 35°)/(tan 55°)`

Evaluate: sin² 40° - cos² 50°

Evaluate: sec² 18° - cosec² 72°

Evaluate: sin 15° cos 15° - cos 75° sin 75°

Evaluate: sin 42° sin 48° - cos 42° cos 48°

Evaluate: sin (90° - A) sin A - cos (90° - A) cos A

Evaluate: sin² 35° - cos² 55°

Evaluate: `(cot54°)/(tan36°) + (tan20°)/(cot70°)-2`

Evaluate: cos² 25° - sin² 65° - tan² 45°

Evaluate: `((sin 77°)/(cos 13°))^2 + ((cos 77°)/(sin 13°))^2 - 2 cos^2 45°`

Show that: tan 10° tan 15° tan 75° tan 80° = 1.

Show that: sin 42° sec 48° + cos 42° cosec 48° = 2.

Express the following in term of angles between 0° and 45° :

sin 59° + tan 63°

Express the following in term of angles between 0° and 45° :

cosec 68° + cot 72°

Express the following in term of angles between 0° and 45° :

cos 74° + sec 67°

For triangle ABC, show that: `sin  (A + B)/2 = cos  C/2`

For triangle ABC, show that : `tan  (B + C)/2 = cot  A/2`

Evaluate: `3(sin72°)/(cos18°) - (sec32°)/("cosec"58°)`.

Evaluate:

3 cos 80° cosec 10°+ 2 sin 59° sec 31°

Evaluate: `(sin 80°)/(cos 10°)`+ sin 59° sec 31°

Evaluate:

tan (55° – A) – cot (35° + A)

Evaluate: cosec (65° + A) - sec (25° - A)

Evaluate: `2(tan57°)/(cot33°) - (cot70°)/(tan20°) - sqrt(2) cos 45°`

Evaluate: `(cot^2 41°)/(tan^2 49°) - 2 (sin^2 75°)/(cos^2 15°)`

Evaluate: `(cos70°)/(sin20°) + (cos59°)/(sin 31°) - 8 sin^2 30°`

Evaluate: 14 sin 30°+ 6 cos 60°- 5 tan 45°.

A triangle ABC is right-angled at B; find the value of `(sec "A". sin "C" - tan "A". tan "C")/sin "B"`.

In the case, given below, find the value of angle A, where 0° ≤ A ≤ 90°.
sin (90° - 3A).cosec 42° = 1.

In the case, given below, find the value of angle A, where 0° ≤ A ≤ 90°.

cos(90° - A) · sec 77° = 1

The value of tan A is ______.

The value of sin B − cos A is ______.

The value of cos 60° − sin 90° + 2 cos 0° is ______.

The value of sin 23° − cos 67° is ______.

Statement (1): The angle C of a right-angled triangle is 90°; then tan A = cot B.

Statement (2): Since angle C of triangle ABC = 90°. ∴ ∠A + ∠B = 90° ⇒ ∠A = 90° − ∠B.

Statement (1): If 4 cos A = 3, sec A = `4/3`.

Statement (2): 4 cos A = 3 ⇒ cos A = `3/4` and sec A = `1/(cos A) = 4/3`.

Assertion (A): The value of sin² 30° − 2 cos³ 60° + 2 tan⁴ 45° is 2.

Reason (R): sin 30° = `1/2`, cos 60° = `1/2`, and tan 45° = 1.

Assertion (A): If A = 30°, the value of 4 sin A sin (60° − A) sin (60° + A) = 1.

Reason (R): 60° − A = 30° and 60° + A = 90°.

If cosec θ = `sqrt5`, find the value of:

1. 2 - sin² θ - cos² θ
2. 2 + `1/sin^2"θ" – cos^2"θ"/sin^2"θ"` 

In the given figure; ∠C = 90^o and D is mid-point of AC.
Find : 
(i) `(tan∠CAB)/ (tan∠CDB)` (ii) `(tan∠ABC)/ (tan∠DBC)`

If 3 cos A = 4 sin A, find the value of :
(i) cos A(ii) 3 - cot² A + cosec²A.

Use the information given in the following figure to evaluate:

`(10)/sin x + (6)/sin y  –  6 cot y`.

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°.

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

For the triangle ABC, show that `sin^2  A/2 + sin^2  (B + C)/2 = 1`.

If seс (90° − 3A) . cos 48° = 1 and 0 ≤ 3A ≤ 90°; find the value of angle A.

In ΔABC, angle C is 90° then find the value of sin (A + B).

If sec A sin A = 0, find the value of cos A.

Find angle A, if sec 2A = cosec (A + 48°).

The shadow of a lamp post is `3sqrt3` m long when the angle of elevation of the sun is 60°. Arushi is of height 3 m standing in front on the lamp post.

1. What is the height of the lamp post?
2. What is the length of shadow of Arushi?
3. How far is Aarushi from the lamp post?