RD Sharma solutions for Mathematics [English] Class 10 chapter 10 - Trigonometric Ratios [Latest edition]

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`sin A = 2/3`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`cos A = 4/5`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

tan θ = 11

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`sin theta = 11/5`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

`tan alpha = 5/12`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`sin theta = sqrt3/2`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`cos theta = 7/25`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

`tan theta = 8/15`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

`cot theta = 12/5`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`sec theta = 13/5`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`cosec theta = sqrt10`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

`cos theta = 12/2`

In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine:

sin A, cos A

In ΔABC right angled at B, AB = 24 cm, BC = 7 m. Determine:

sin C, cos C

In Fig below, Find tan P and cot R. Is tan P = cot R?

If `sin A = 9/41` compute cos 𝐴 𝑎𝑛𝑑 tan 𝐴

Given 15 cot A = 8. Find sin A and sec A.

In ΔPQR, right angled at Q, PQ = 4 cm and RQ = 3 cm. Find the values of sin P, sin R, sec P and sec R.

If cot θ =` 7/8` evaluate `((1+sin θ )(1-sin θ))/((1+cos θ)(1-cos θ))`

If cot θ = `7/8`, evaluate cot² θ.

If 3 cot A = 4, Check whether `((1-tan^2 A)/(1+tan^2 A)) = cos^2 "A" - sin^2 "A"` or not.

If `tan theta = a/b`, find the value of `(cos theta + sin theta)/(cos theta - sin theta)`

If 3 tan θ = 4, find the value of `(4cos theta - sin theta)/(2cos theta + sin theta)`

If 3 cot θ = 2, find the value of  `(4sin theta - 3 cos theta)/(2 sin theta + 6cos theta)`.

If tan θ = `a/b` prove that `(a sin theta - b cos theta)/(a sin theta + b cos theta) = (a^2 - b^2)/(a^2 + b^2)`

If sec θ = `13/5, "show that"  (2sinθ - 3 cosθ)/(4sinθ - 9cosθ) = 3`.

If `cos theta = 12/13`, show that `sin theta (1 - tan theta) = 35/156`

If `cot theta = 1/sqrt3` show that  `(1 - cos^2 theta)/(2 - sin^2  theta) = 3/5`

If `tan theta = 1/sqrt7`     `(cosec^2 theta - sec^2 theta)/(cosec^2 theta + sec^2 theta) = 3/4`

If sin θ = `12/13`, Find `(sin^2 θ - cos^2 θ)/(2sin θ cos θ) × 1/(tan^2 θ)`.

if `sec theta = 5/4` find the value of `(sin theta - 2 cos theta)/(tan theta - cot theta)`

if `cos theta = 5/13` find the value of `(sin^2 theta - cos^2 theta)/(2 sin theta cos theta) = 3/5`

if `tan theta = 12/13` Find `(2 sin theta cos theta)/(cos^2 theta - sin^2 theta)`

if `cos theta = 3/5`, find the value of `(sin theta - 1/(tan theta))/(2 tan theta)`

if `sin theta = 3/5  " evaluate " (cos theta - 1/(tan theta))/(2 cot theta)`

 if `sec A = 5/4` verify that `(3 sin A - 4 sin^3 A)/(4 cos^3 A - 3 cos A) = (3 tan A - tan^3 A)/(1- 3 tan^2 A)`

if `sin theta = 3/4`  prove that `sqrt(cosec^2 theta - cot)/(sec^2 theta - 1) = sqrt7/3`

if `sec A = 17/8` verify that `(3 - 4sin^2A)/(4 cos^2 A - 3) = (3 - tan^2 A)/(1 - 3 tan^2 A)`

if `cot theta = 3/4` prove that `sqrt((sec theta - cosec theta)/(sec theta +cosec theta)) = 1/sqrt7`

If `tan theta = 24/7`, find that sin 𝜃 + cos 𝜃

If `sin theta = a/b` find sec θ + tan θ in terms of a and b.

If 8 tan A = 15, find sin A – cos A.

If 3cos θ – 4sin  = 2cos θ + sin θ Find tan θ.

If `tan θ = 20/21` show that `(1 - sin theta + cos theta)/(1 + sin theta + cos theta) = 3/7`

If Cosec A = 2 find `1/(tan A) + (sin A)/(1 + cos A)`

If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.

If ∠A and ∠P are acute angles such that tan A = tan P, then show that ∠A = ∠P.

In a ΔABC, right angled at A, if tan C = `sqrt3` , find the value of sin B cos C + cos B sin C.

State whether the following are true or false. Justify your answer.

The value of tan A is always less than 1.

State whether the following are true or false. Justify your answer.

sec A = `12/5` for some value of angle A.

State whether the following are true or false. Justify your answer.

cos A is the abbreviation used for the cosecant of angle A.

State whether the following are true or false. Justify your answer.

sin θ = `4/3`, for some angle θ.

Evaluate the following

sin 45° sin 30° + cos 45° cos 30°

Evaluate the following in the simplest form:

sin 60° cos 30° + cos 60° sin 30°

Evaluate the following

cos 60° cos 45° - sin 60° ∙ sin 45°

Evaluate the following

sin² 30° + sin² 45° + sin² 60° + sin² 90°

Evaluate the following

cos² 30° + cos² 45° + cos² 60° + cos² 90°

Evaluate the following

tan² 30° + tan² 60° + tan² 45°

Evaluate the following

`2 sin^2 30^2 - 3 cos^2 45^2 + tan^2 60^@`

Evaluate the following

`sin^2 30° cos^2 45 ° + 4 tan^2 30° + 1/2 sin^2 90° − 2 cos^2 90° + 1/24 cos^2 0°`

Evaluate the Following

4(sin⁴ 60° + cos⁴ 30°) − 3(tan² 60° − tan² 45°) + 5 cos² 45°

Evaluate the following:

(cosec² 45° sec² 30°)(sin² 30° + 4 cot² 45° − sec² 60°)

Evaluate the Following

cosec³ 30° cos 60° tan3 45° sin² 90° sec² 45° cot 30°

Evaluate the Following

`cot^2 30^@ - 2 cos^2 60^circ- 3/4 sec^2 45^@ - 4 sec^2 30^@`

Evaluate the Following

(cos 0° + sin 45° + sin 30°)(sin 90° − cos 45° + cos 60°)

Evaluate the Following

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

Evaluate the Following

`4/(cot^2 30^@) + 1/(sin^2 60^@) - cos^2 45^@`

Evaluate the Following

4(sin⁴ 30° + cos² 60°) − 3(cos² 45° − sin² 90°) − sin² 60°

Evaluate the Following

`(tan^2 60^@ + 4 cos^2 45^@ + 3 sec^2 30^@ + 5 cos^2 90)/(cosec 30^@ + sec 60^@ - cot^2 30^@)`

Evaluate the Following

`sin 30^2/sin 45^@ + tan 45^@/sec 60^@ - sin 60^@/cot 45^@ - cos 30^@/sin 90^@`

Evaluate the Following:

`tan 45^@/(cosec 30^@) + sec 60^@/cot 45^@  - (5 sin 90^@)/(2 cos 0^@)`

Find the value of x in the following :

`2sin 3x = sqrt3`

Find the value of x in the following :

`2 sin  x/2 = 1`

Find the value of x in the following :

`sqrt3 sin x = cos x`

Find the value of x in the following :

tan 3x = sin 45º cos 45º + sin 30º

Find the value of x in the following :

`sqrt3 tan 2x = cos 60^@ + sin45^@ cos 45^@`

Find the value of x in the following :

cos 2x = cos 60° cos 30° + sin 60° sin 30°

If θ = 30° verify `tan 2 theta = (2 tan theta)/(1 - tan^2 theta)`

If θ = 30° verify that  `sin 2theta = (2 tan theta)/(1 + tan^2 theta)`

If 𝜃 = 30° verify `cos 2 theta = (1 - tan^2 theta)/(1 + tan^2 theta)`

f θ = 30°, verify that cos 3θ = 4 cos³ θ − 3 cos θ

If A = B = 60°, verify that cos (A − B) = cos A cos B + sin A sin B

If A = B = 60°, verify that sin (A − B) = sin A cos B − cos A sin B

If A = B = 60°. Verify `tan (A - B) = (tan A - tan B)/(1 + tan   tan B)`

If A = 30° B = 60° verify 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

If sin (A − B) = sin A cos B − cos A sin B and cos (A − B) = cos A cos B + sin A sin B, find the values of sin 15° and cos 15°.

In right angled triangle ABC. ∠C = 90°, ∠B = 60°. AB = 15 units. Find remaining angles and sides.

In ΔABC is a right triangle such that ∠C = 90° ∠A = 45°, BC = 7 units find ∠B, AB and AC

In rectangle ABCD AB = 20cm ∠BAC = 60° BC, calculate side BC and diagonals AC and BD.

If Sin (A + B) = 1 and cos (A – B) = 1, 0° < A + B ≤ 90° A ≥ B. Find A & B

If tan (A + B) = `sqrt3` and tan (A – B) = `1/sqrt3`; 0° < A + B ≤ 90°; A > B, find A and B.

If `sin (A – B) = 1/2` and `cos (A + B) = 1/2`, `0^@` < A + `B <= 90^@`, A > B Find A and B.

In right angled triangle ΔABC at B, ∠A = ∠C. Find the values of Sin A cos C + Cos A Sin C

In right angled triangle ΔABC at B, ∠A = ∠C. Find the values of sin A sin B + cos A cos B

Find acute angles A & B, if sin (A + 2B) = `sqrt3/2 cos(A + 4B) = 0, A > B`

If A and B are acute angles such that tan A = 1/2,  tan B = 1/3 and tan (A + B) = `(tan A + tan B)/(1- tan A tan B)` A + B = ?

In ∆PQR, right-angled at Q, PQ = 3 cm and PR = 6 cm. Determine ∠P and ∠R.

Evaluate the following:

`(sin 20^@)/(cos 70^@)`

Evaluate the following :

`cos 19^@/sin 71^@`

Evaluate the following :

`(sin 21^@)/(cos 69^@)`

Evaluate the following :

`tan 10^@/cot 80^@`

Evaluate the following

`sec 11^@/(cosec 79^@)`

Evaluate the following :

`((sin 49^@)/(cos 41^@))^2 + (cos 41^@/(sin 49^@))^2`

Evaluate cos 48° − sin 42°

Evaluate the following :

`(cot 40^@)/cos 35^@ -  1/2 [(cos 35^@)/(sin 55^@)]`

Evaluate the following :

`((sin 27^@)/(cos 63^@))^2 - (cos 63^@/sin 27^@)^2`

Evaluate the following :

`tan 35^@/cot 55^@  + cot 78^@/tan 12^@  -1`

Evaluate the following :

`(sec 70^@)/(cosec 20^@) + (sin 59^@)/(cos 31^@)`

Evaluate the following :

cosec 31° − sec 59°

Evaluate the following :

(sin 72° + cos 18°) (sin 72° − cos 18°)

Evaluate the following :

sin 35° sin 55° − cos 35° cos 55°

Show that tan 48° tan 23° tan 42° tan 67° = 1

Evaluate the following 

sec 50º sin 40° + cos 40º cosec 50º 

Express each one of the following in terms of trigonometric ratios of angles lying between
0° and 45°

Sin 59° + cos 56°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

tan 65° + cot 49°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

sec 76° + cosec 52°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

cos 78° + sec 78°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

cosec 54° + sin 72°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

cot 85° + cos 75°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

 sin 67° + cos 75°

Express cos 75° + cot 75° in terms of angles between 0° and 30°.

If Sin 3A = cos (A – 26°), where 3A is an acute angle, find the value of A =?

If A, B, C are the interior angles of a triangle ABC, prove that

`tan ((C+A)/2) = cot  B/2`

If A, B and C are interior angles of a triangle ABC, then show that `\sin( \frac{B+C}{2} )=\cos \frac{A}{2}`

Prove that  tan 20° tan 35° tan 45° tan 55° tan 70° = 1

Prove that sin 48° sec 42° + cos 48° cosec 42° = 2

Prove that `sin 70^@/cos 20^@  + (cosec 20^@)/sec 70^@  -  2 cos 20^@ cosec 20^@ = 0`

Prove that `cos 80^@/sin 10^@  + cos 59^@ cosec 31^@ = 2`

Prove the following

sin θ sin (90° − θ) − cos θ cos (90° − θ) = 0

Prove the following :

`(cos(90^@ - theta) sec(90^@ - theta)tan theta)/(cosec(90^@ - theta) sin(90^@ - theta) cot (90^@ -  theta)) + tan (90^@ - theta)/cot theta = 2`

Prove the following

`(tan (90 - A) cot A)/(cosec^2 A)   - cos^2 A =0`

Prove the following :

`(cos(90°−A) sin(90°−A))/tan(90°−A) - sin^2 A = 0`

Prove the following

 sin (50° − θ) − cos (40° − θ) + tan 1° tan 10° tan 20° tan 70° tan 80° tan 89° = 1

Evaluate:

`2/3 (cos^4 30° - sin^4 45°) - 3(sin^2 60° - sec^2 45°) + 1/4 cot^2 30°`.

Evaluate: `4(sin^2 30 + cos^4 60^@) - 2/3 3[(sqrt(3/2))^2 . [1/sqrt2]^2] + 1/4 (sqrt3)^2`

Evaluate: `sin 50^@/cos 40^@ + (cosec 40^@)/sec 50^@  - 4 cos 50^@ cosec 40^@`

Evaluate tan 35° tan 40° tan 50° tan 55°

Evaluate: Cosec (65 + θ) – sec (25 – θ) – tan (55 – θ) + cot (35 + θ)

Evaluate: tan 7° tan 23° tan 60° tan 67° tan 83°

Evaluate: `(2sin 68)/cos 22 - (2 cot 15^@)/(5 tan 75^@) - (8 tan 45^@ tan 20^@ tan 40^@ tan 50^@ tan 70^@)/5`

Evaluate: `(3 cos 55^@)/(7 sin 35^@) -  (4(cos 70 cosec 20^@))/(7(tan 5^@ tan 25^@ tan 45^@ tan 65^@ tan  85^@))`

Evaluate: `sin 18^@/cos 72^@  + sqrt3 [tan 10° tan 30° tan 40° tan 50° tan 80°]`

Evaluate: `cos 58^@/sin 32^@  + sin 22^@/cos 68^@  - (cos 38^@ cosec 52^@)/(tan 18^@ tan 35^@ tan 60^@ tan 72^@ tan 65^@)`

If sin θ = cos (θ – 45°), where θ – 45° are acute angles, find the degree measure of θ

If A, B, C are the interior angles of a ΔABC, show that `cos[(B+C)/2] = sin A/2`

If 2θ + 45° and 30° − θ are acute angles, find the degree measure of θ satisfying Sin (20 + 45°) = cos (30 - θ°)

If θ is a positive acute angle such that sec θ = cosec 60°, find 2 cos2 θ – 1

If cos 2θ = sin 4θ where 2θ, 4θ are acute angles, find the value of θ.

If sin 3θ = cos (θ – 6°) where 3θ and θ − 6° are acute angles, find the value of θ.

If Sec 4A = cosec (A – 20°) where 4A is an acute angle, find the value of A.

If sec 2A = cosec (A – 42°) where 2A is an acute angle. Find the value of A.

Write the maximum and minimum values of sin θ.

Write the maximum and minimum values of cos θ.

What is the maximum value of \[\frac{1}{\sec \theta}\] 

What is the maximum value of \[\frac{1}{\sec \theta}\]

If \[\tan \theta = \frac{4}{5}\] find the value of \[\frac{\cos \theta - \sin \theta}{\cos \theta + \sin \theta}\]

If \[\cos \theta = \frac{2}{3}\]  find the value of \[\frac{\sec \theta - 1}{\sec \theta + 1}\]

If 3 cot θ = 4, find the value of \[\frac{4 \cos \theta - \sin \theta}{2 \cos \theta + \sin \theta}\]

Given 

\[\frac{4 \cos \theta - \sin \theta}{2 \cos \theta + \sin \theta}\] what is the value of \[\frac{{cosec}^2 \theta - \sec^2 \theta}{{cosec}^2 \theta + \sec^2 \theta}\]

If \[\frac{{cosec}^2 \theta - \sec^2 \theta}{{cosec}^2 \theta + \sec^2 \theta}\]  write the value of \[\frac{1 - \cos^2 \theta}{2 - \sin^2 \theta}\] 

If \[\tan A = \frac{3}{4} \text{ and } A + B = 90°\] then what is the value of cot B?

If A + B = 90° and \[\cos B = \frac{3}{5}\]  what is the value of sin A? 

Write the acute angle θ satisfying \[\cos B = \frac{3}{5}\]

Write the value of cos 1° cos 2° cos 3° ....... cos 179° cos 180°. 

Write the value of tan 10° tan 15° tan 75° tan 80°?

If A + B = 90° and \[\tan A = \frac{3}{4}\]\[\tan A = \frac{3}{4}\] what is cot B? 

If \[\tan A = \frac{5}{12}\] \[\tan A = \frac{5}{12}\]  find the value of (sin A + cos A) sec A. 

If θ is an acute angle such that \[\cos \theta = \frac{3}{5}, \text{ then } \frac{\sin \theta \tan \theta - 1}{2 \tan^2 \theta} =\] \[\cos \theta = \frac{3}{5}, \text{ then } \frac{\sin \theta \tan \theta - 1}{2 \tan^2 \theta} =\] 

If \[\frac{160}{3}\] \[\tan \theta = \frac{a}{b}, \text{ then } \frac{a \sin \theta + b \cos \theta}{a \sin \theta - b \cos \theta}\]

If 5 tan θ − 4 = 0, then the value of \[\frac{5 \sin \theta - 4 \cos \theta}{5 \sin \theta + 4 \cos \theta}\] is:

If 16 cot x = 12, then \[\frac{\sin x - \cos x}{\sin x + \cos x}\]

If 8 tan x = 15, then sin x − cos x is equal to 

If \[\tan \theta = \frac{1}{\sqrt{7}}, \text{ then } \frac{{cosec}^2 \theta - \sec^2 \theta}{{cosec}^2 \theta + \sec^2 \theta} =\] 

If \[\tan \theta = \frac{3}{4}\]  then cos² θ − sin² θ = 

If θ is an acute angle such that \[\tan^2 \theta = \frac{8}{7}\] then the value of \[\frac{\left( 1 + \sin \theta \right) \left( 1 - \sin \theta \right)}{\left( 1 + \cos \theta \right) \left( 1 - \cos \theta \right)}\]

If 3 cos θ = 5 sin θ, then the value of

If tan² 45° − cos² 30° = x sin 45° cos 45°, then x = 

The value of cos² 17° − sin² 73° is 

The value of \[\frac{\cos^3 20°- \cos^3 70°}{\sin^3 70° - \sin^3 20°}\] 

If \[\frac{x {cosec}^2 30°\sec^2 45°}{8 \cos^2 45° \sin^2 60°} = \tan^2 60° - \tan^2 30°\] 

If A and B are complementary angles, then

If x sin (90° − θ) cot (90° − θ) = cos (90° − θ), then x =

If x tan 45° cos 60° = sin 60° cot 60°, then x is equal to 

If angles A, B, C to a ∆ABC from an increasing AP, then sin B = 

If θ is an acute angle such that sec² θ = 3, then the value of \[\frac{\tan^2 \theta - {cosec}^2 \theta}{\tan^2 \theta + {cosec}^2 \theta}\]

The value of tan 1° tan 2° tan 3° ...... tan 89° is 

The value of cos 1° cos 2° cos 3° ..... cos 180° is 

The value of tan 10° tan 15° tan 75° tan 80° is 

The value of

If θ and 2θ − 45° are acute angles such that sin θ = cos (2θ − 45°), then tan θ is equal to 

If A + B = 90°, then \[\frac{\tan A \tan B + \tan A \cot B}{\sin A \sec B} - \frac{\sin^2 B}{\cos^2 A}\] 

If 5θ and 4θ are acute angles satisfying sin 5θ = cos 4θ, then 2 sin 3θ −\[\sqrt{3} \tan 3\theta\]  is equal to 

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

\[\frac{1 - \tan^2 45°}{1 + \tan^2 45°}\] is equal to 

Sin 2A = 2 sin A is true when A =

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

If A, B and C are interior angles of a triangle ABC, then \[\sin \left( \frac{B + C}{2} \right) =\]

If \[\cos \theta = \frac{2}{3}\]  then 2 sec² θ + 2 tan² θ − 7 is equal to 

tan 5° ✕ tan 30° ✕ 4 tan 85° is equal to 

The value of \[\frac{\tan 55°}{\cot 35°}\] + cot 1° cot 2° cot 3° .... cot 90°, is

In the following figure  the value of cos ϕ is 

In the following Figure. AD = 4 cm, BD = 3 cm and CB = 12 cm, find the cot θ.