Share

# RD Sharma solutions for Class 10 Mathematics chapter 10 - Trigonometric Ratios

## Chapter 10: Trigonometric Ratios

Ex. 10.10Ex. 10.20Ex. 10.30Others

#### Chapter 10: Trigonometric Ratios Exercise 10.10 solutions [Pages 23 - 26]

Ex. 10.10 | Q 1.01 | Page 23

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

sin A = 2/3

Ex. 10.10 | Q 1.02 | Page 23

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

cos A = 4/5

Ex. 10.10 | Q 1.03 | Page 23

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

tan θ = 11

Ex. 10.10 | Q 1.04 | Page 23

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

sin theta = 11/5

Ex. 10.10 | Q 1.05 | Page 23

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

tan alpha = 5/12

Ex. 10.10 | Q 1.06 | Page 23

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

sin theta = sqrt3/2

Ex. 10.10 | Q 1.07 | Page 23

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

cos theta = 7/25

Ex. 10.10 | Q 1.08 | Page 23

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

tan theta = 8/15

Ex. 10.10 | Q 1.09 | Page 23

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

cot theta = 12/5

Ex. 10.10 | Q 1.1 | Page 23

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

sec theta = 13/5

Ex. 10.10 | Q 1.11 | Page 23

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

cosec theta = sqrt10

Ex. 10.10 | Q 1.12 | Page 23

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

cos theta = 12/2

Ex. 10.10 | Q 2.1 | Page 23

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

Sin A, Cos A

Ex. 10.10 | Q 2.2 | Page 23

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

Sin C, cos C

Ex. 10.10 | Q 3 | Page 23

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

Ex. 10.10 | Q 4 | Page 24

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

Ex. 10.10 | Q 5 | Page 24

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

Ex. 10.10 | Q 6 | Page 24

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.

Ex. 10.10 | Q 7.1 | Page 24

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

Ex. 10.10 | Q 7.2 | Page 24

If cot θ = 7/8, evaluate cot2 θ

Ex. 10.10 | Q 8 | Page 24

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

Ex. 10.10 | Q 9 | Page 24

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

Ex. 10.10 | Q 10 | Page 24

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

Ex. 10.10 | Q 11 | Page 24

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

Ex. 10.10 | Q 12 | Page 24

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)

Ex. 10.10 | Q 13 | Page 24

if sec theta = 13/5 show that (2 cos theta - 3 cos theta)/(4 sin theta - 9 cos theta) = 3

Ex. 10.10 | Q 14 | Page 24

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

Ex. 10.10 | Q 15 | Page 24

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

Ex. 10.10 | Q 16 | Page 24

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

Ex. 10.10 | Q 17 | Page 24

if sin theta = 12/13 find (sin^2 theta - cos^2 theta)/(2sin theta cos theta) xx 1/(tan^2 theta)

Ex. 10.10 | Q 18 | Page 24

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

Ex. 10.10 | Q 19 | Page 25

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

Ex. 10.10 | Q 20 | Page 25

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

Ex. 10.10 | Q 21 | Page 25

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

Ex. 10.10 | Q 22 | Page 25

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

Ex. 10.10 | Q 23 | Page 25

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)

Ex. 10.10 | Q 24 | Page 25

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

Ex. 10.10 | Q 25 | Page 25

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

Ex. 10.10 | Q 26 | Page 25

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

Ex. 10.10 | Q 27 | Page 25

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

Ex. 10.10 | Q 28 | Page 25

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

Ex. 10.10 | Q 29 | Page 25

If 8 tan A = 15, find sin A – cos A.

Ex. 10.10 | Q 30 | Page 25

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

Ex. 10.10 | Q 31 | Page 25

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

Ex. 10.10 | Q 32 | Page 25

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

Ex. 10.10 | Q 33 | Page 25

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

Ex. 10.10 | Q 34 | Page 26

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

Ex. 10.10 | Q 35 | Page 26

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

Ex. 10.10 | Q 36.1 | Page 26

The value of tan A is always less than 1.

Ex. 10.10 | Q 36.2 | Page 26

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

Ex. 10.10 | Q 36.3 | Page 26

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

Ex. 10.10 | Q 36.4 | Page 26

sin θ =4/3, for some angle θ

#### Chapter 10: Trigonometric Ratios Exercise 10.20 solutions [Pages 41 - 43]

Ex. 10.20 | Q 1 | Page 41

Evaluate the following

sin 45° sin 30° + cos 45° cos 30°

Ex. 10.20 | Q 2 | Page 41

Evaluate the following

sin 60° cos 30° + cos 60° sin 30°

Ex. 10.20 | Q 3 | Page 41

Evaluate the following

cos 60° cos 45° - sin 60° ∙ sin 45°

Ex. 10.20 | Q 4 | Page 41

Evaluate the following

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

Ex. 10.20 | Q 5 | Page 41

Evaluate the following

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

Ex. 10.20 | Q 6 | Page 41

Evaluate the following

tan2 30° + tan2 60° + tan45°

Ex. 10.20 | Q 7 | Page 41

Evaluate the following

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

Ex. 10.20 | Q 8 | Page 41

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°

Ex. 10.20 | Q 9 | Page 42

Evaluate the Following

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

Ex. 10.20 | Q 10 | Page 42

Evaluate the following

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

Ex. 10.20 | Q 11 | Page 42

Evaluate the Following

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

Ex. 10.20 | Q 12 | Page 42

Evaluate the Following

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

Ex. 10.20 | Q 13 | Page 42

Evaluate the Following

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

Ex. 10.20 | Q 14 | Page 42

Evaluate the Following

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

Ex. 10.20 | Q 15 | Page 42

Evaluate the Following

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

Ex. 10.20 | Q 16 | Page 42

Evaluate the Following

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

Ex. 10.20 | Q 17 | Page 42

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^@)

Ex. 10.20 | Q 18 | Page 42

Evaluate the Following

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

Ex. 10.20 | Q 19 | Page 42

Evaluate the Following

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

Ex. 10.20 | Q 20 | Page 42

Find the value of x in the following :

2sin 3x = sqrt3

Ex. 10.20 | Q 21 | Page 42

Find the value of x in the following :

2 sin  x/2 = 1

Ex. 10.20 | Q 22 | Page 42

Find the value of x in the following :

sqrt3 sin x = cos x

Ex. 10.20 | Q 23 | Page 42

Find the value of x in the following :

tan 3x = sin 45º cos 45º + sin 30º

Ex. 10.20 | Q 24 | Page 42

Find the value of x in the following :

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

Ex. 10.20 | Q 25 | Page 42

Find the value of x in the following :

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

Ex. 10.20 | Q 26.1 | Page 42

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

Ex. 10.20 | Q 26.2 | Page 42

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

Ex. 10.20 | Q 26.3 | Page 42

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

Ex. 10.20 | Q 26.4 | Page 42

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

Ex. 10.20 | Q 27.1 | Page 42

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

Ex. 10.20 | Q 27.2 | Page 42

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

Ex. 10.20 | Q 27.3 | Page 42

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

Ex. 10.20 | Q 28.1 | Page 42

If A = 30° B = 60° verify Sin (A + B) = Sin A Cos B + cos A sin B

Ex. 10.20 | Q 28.2 | Page 42

If A = 30° and B = 60°, verify that cos (A + B) = cos A cos B − sin A sin B

Ex. 10.20 | Q 29 | Page 43

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°.

Ex. 10.20 | Q 30 | Page 43

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

Ex. 10.20 | Q 31 | Page 43

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

Ex. 10.20 | Q 32 | Page 43

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

Ex. 10.20 | Q 33 | Page 43

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

Ex. 10.20 | Q 34 | Page 43

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

Ex. 10.20 | Q 35 | Page 43

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

Ex. 10.20 | Q 36.1 | Page 43

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

Ex. 10.20 | Q 36.2 | Page 43

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

Ex. 10.20 | Q 37 | Page 43

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

Ex. 10.20 | Q 38 | Page 43

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 = ?

Ex. 10.20 | Q 39 | Page 43

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

#### Chapter 10: Trigonometric Ratios Exercise 10.30 solutions [Pages 52 - 54]

Ex. 10.30 | Q 1.1 | Page 52

Evaluate the following:

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

Ex. 10.30 | Q 1.2 | Page 52

Evaluate the following :

cos 19^@/sin 71^@

Ex. 10.30 | Q 1.3 | Page 52

Evaluate the following :

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

Ex. 10.30 | Q 1.4 | Page 52

Evaluate the following :

tan 10^@/cot 80^@

Ex. 10.30 | Q 1.5 | Page 52

Evaluate the following

sec 11^@/(cosec 79^@)

Ex. 10.30 | Q 2.01 | Page 53

Evaluate the following :

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

Ex. 10.30 | Q 2.02 | Page 53

Evaluate cos 48° − sin 42°

Ex. 10.30 | Q 2.03 | Page 53

Evaluate the following :

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

Ex. 10.30 | Q 2.04 | Page 53

Evaluate the following :

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

Ex. 10.30 | Q 2.05 | Page 53

Evaluate the following :

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

Ex. 10.30 | Q 2.06 | Page 53

Evaluate the following :

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

Ex. 10.30 | Q 2.07 | Page 53

Evaluate the following :

cosec 31° − sec 59°

Ex. 10.30 | Q 2.08 | Page 53

Evaluate the following :

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

Ex. 10.30 | Q 2.09 | Page 53

Evaluate the following :

sin 35° sin 55° − cos 35° cos 55°

Ex. 10.30 | Q 2.1 | Page 53

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

Ex. 10.30 | Q 2.11 | Page 53

Evaluate the following

sec 50º sin 40° + cos 40º cosec 50º

Ex. 10.30 | Q 3.1 | Page 53

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

Sin 59° + cos 56°

Ex. 10.30 | Q 3.2 | Page 53

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

tan 65° + cot 49°

Ex. 10.30 | Q 3.3 | Page 53

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

sec 76° + cosec 52°

Ex. 10.30 | Q 3.4 | Page 53

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

cos 78° + sec 78°

Ex. 10.30 | Q 3.5 | Page 53

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

cosec 54° + sin 72°

Ex. 10.30 | Q 3.6 | Page 53

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

cot 85° + cos 75°

Ex. 10.30 | Q 3.7 | Page 53

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

sin 67° + cos 75°

Ex. 10.30 | Q 4 | Page 53

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

Ex. 10.30 | Q 5 | Page 53

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

Ex. 10.30 | Q 6.1 | Page 53

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

tan ((C+A)/2) = cot  B/2

Ex. 10.30 | Q 6.2 | Page 53

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

Ex. 10.30 | Q 7.1 | Page 53

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

Ex. 10.30 | Q 7.2 | Page 53

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

Ex. 10.30 | Q 7.3 | Page 53

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

Ex. 10.30 | Q 7.4 | Page 53

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

Ex. 10.30 | Q 8.1 | Page 53

Prove the following

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

Ex. 10.30 | Q 8.2 | Page 53

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

Ex. 10.30 | Q 8.3 | Page 53

Prove the following

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

Ex. 10.30 | Q 8.4 | Page 53

Prove the following :

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

Ex. 10.30 | Q 8.5 | Page 53

Prove the following

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

Ex. 10.30 | Q 9.01 | Page 54

Evaluate: 2/3 (cos^4 30^@ - sin^4 45^@) - 3(sin^2 60^@ - sec^2 45^@) + 1/4 cot^2 30^@

Ex. 10.30 | Q 9.02 | Page 54

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

Ex. 10.30 | Q 9.03 | Page 54

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

Ex. 10.30 | Q 9.04 | Page 54

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

Ex. 10.30 | Q 9.05 | Page 54

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

Ex. 10.30 | Q 9.06 | Page 54

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

Ex. 10.30 | Q 9.07 | Page 54

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

Ex. 10.30 | Q 9.08 | Page 54

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

Ex. 10.30 | Q 9.09 | Page 54

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

Ex. 10.30 | Q 9.1 | Page 54

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

Ex. 10.30 | Q 10 | Page 54

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

Ex. 10.30 | Q 11 | Page 54

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

Ex. 10.30 | Q 12 | Page 54

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

Ex. 10.30 | Q 13 | Page 54

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

Ex. 10.30 | Q 14 | Page 54

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

Ex. 10.30 | Q 15 | Page 54

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

Ex. 10.30 | Q 16 | Page 54

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

Ex. 10.30 | Q 17 | Page 54

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

#### Chapter 10: Trigonometric Ratios solutions [Pages 55 - 56]

Q 1 | Page 55

Write the maximum and minimum values of sin θ.

Q 1 | Page 55

Write the maximum and minimum values of sin θ.

Q 2 | Page 55

Write the maximum and minimum values of cos θ.

Q 2 | Page 55

Write the maximum and minimum values of cos θ.

Q 3 | Page 55

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

Q 3 | Page 55

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

Q 4 | Page 55

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

Q 4 | Page 55

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

Q 5 | Page 55

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

Q 5 | Page 55

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

Q 6 | Page 55

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

Q 6 | Page 55

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

Q 7 | Page 55

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

Q 7 | Page 55

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

Q 8 | Page 55

Given

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

Q 8 | Page 55

Given

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

Q 9 | Page 55

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

Q 9 | Page 55

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

Q 10 | Page 55

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

Q 10 | Page 55

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

Q 11 | Page 56

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

Q 11 | Page 56

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

Q 12 | Page 56

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

Q 12 | Page 56

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

Q 13 | Page 55

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

Q 13 | Page 55

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

Q 14 | Page 56

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

Q 14 | Page 56

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

Q 15 | Page 55

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

Q 15 | Page 55

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

Q 16 | Page 55

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

Q 16 | Page 55

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

#### Chapter 10: Trigonometric Ratios solutions [Pages 56 - 59]

Q 1 | Page 56

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

• $\frac{16}{625}$

• $\frac{1}{36}$

• $\frac{3}{160}$

• $\frac{160}{3}$

Q 1 | Page 56

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

• $\frac{16}{625}$

• $\frac{1}{36}$

• $\frac{3}{160}$

• $\frac{160}{3}$

Q 2 | Page 56

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

• $\frac{a^2 + b^2}{a^2 - b^2}$

• $\frac{a^2 - b^2}{a^2 + b^2}$

• $\frac{a + b}{a - b}$

• $\frac{a - b}{a + b}$

Q 2 | Page 56

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

• $\frac{a^2 + b^2}{a^2 - b^2}$

• $\frac{a^2 - b^2}{a^2 + b^2}$

• $\frac{a + b}{a - b}$

• $\frac{a - b}{a + b}$

Q 3 | Page 56

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

• $\frac{5}{3}$

• $\frac{5}{6}$

•  0

• $\frac{1}{6}$

Q 3 | Page 56

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

• $\frac{5}{3}$

• $\frac{5}{6}$

•  0

• $\frac{1}{6}$

Q 4 | Page 56

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

• $\frac{1}{7}$

• $\frac{3}{7}$

• $\frac{2}{7}$

• 0

Q 4 | Page 56

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

• $\frac{1}{7}$

• $\frac{3}{7}$

• $\frac{2}{7}$

• 0

Q 5 | Page 56

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

• $\frac{8}{17}$

• $\frac{17}{7}$

• $\frac{1}{17}$

• $\frac{7}{17}$

Q 5 | Page 56

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

• $\frac{8}{17}$

• $\frac{17}{7}$

• $\frac{1}{17}$

• $\frac{7}{17}$

Q 6 | Page 56

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

• $\frac{5}{7}$

• $\frac{3}{7}$

• $\frac{1}{12}$

• $\frac{3}{4}$

Q 6 | Page 56

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

• $\frac{5}{7}$

• $\frac{3}{7}$

• $\frac{1}{12}$

• $\frac{3}{4}$

Q 7 | Page 57

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

• $\frac{7}{25}$

•  1

• $\frac{- 7}{25}$

• $\frac{4}{25}$

Q 7 | Page 57

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

• $\frac{7}{25}$

•  1

• $\frac{- 7}{25}$

• $\frac{4}{25}$

Q 8 | Page 57

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

• $\frac{7}{8}$

• $\frac{8}{7}$

• $\frac{7}{4}$

• $\frac{64}{49}$

Q 8 | Page 57

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

• $\frac{7}{8}$

• $\frac{8}{7}$

• $\frac{7}{4}$

• $\frac{64}{49}$

Q 9 | Page 57

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

$\frac{5 \sin \theta - 2 \sec^3 \theta + 2 \cos \theta}{5 \sin \theta + 2 \sec^3 \theta - 2 \cos \theta}$

• $\frac{271}{979}$

• $\frac{316}{2937}$

• $\frac{542}{2937}$

• None of these

Q 9 | Page 57

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

$\frac{5 \sin \theta - 2 \sec^3 \theta + 2 \cos \theta}{5 \sin \theta + 2 \sec^3 \theta - 2 \cos \theta}$

• $\frac{271}{979}$

• $\frac{316}{2937}$

• $\frac{542}{2937}$

• None of these

Q 10 | Page 57

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

• 2

•  −2

• $- \frac{1}{2}$

• $\frac{1}{2}$

Q 10 | Page 57

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

• 2

•  −2

• $- \frac{1}{2}$

• $\frac{1}{2}$

Q 11 | Page 57

The value of cos2 17° − sin2 73° is

•  1

• $\frac{1}{3}$

• 0

• -1

Q 11 | Page 57

The value of cos2 17° − sin2 73° is

•  1

• $\frac{1}{3}$

• 0

• -1

Q 12 | Page 57

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

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

•  1

Q 12 | Page 57

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

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

•  1

Q 13 | Page 57

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

•  1

•  −1

•  2

• 0

Q 13 | Page 57

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

•  1

•  −1

•  2

• 0

Q 14 | Page 57

If A and B are complementary angles, then

• sin A = sin B

• cos A = cos B

•  tan A = tan B

• sec A = cosec B

Q 14 | Page 57

If A and B are complementary angles, then

• sin A = sin B

• cos A = cos B

•  tan A = tan B

• sec A = cosec B

Q 15 | Page 57

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

• 0

•  1

•  −1

• 2

Q 15 | Page 57

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

• 0

•  1

•  −1

• 2

Q 16 | Page 57

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

• 1

• $\sqrt{3}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

Q 16 | Page 57

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

• 1

• $\sqrt{3}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{2}}$

Q 17 | Page 57

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

• $\frac{1}{2}$

• $\frac{\sqrt{3}}{2}$

• 1

• $\frac{1}{\sqrt{2}}$

Q 17 | Page 57

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

• $\frac{1}{2}$

• $\frac{\sqrt{3}}{2}$

• 1

• $\frac{1}{\sqrt{2}}$

Q 18 | Page 57

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

• $\frac{4}{7}$

• $\frac{3}{7}$

• $\frac{2}{7}$

• $\frac{1}{7}$

Q 18 | Page 57

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

• $\frac{4}{7}$

• $\frac{3}{7}$

• $\frac{2}{7}$

• $\frac{1}{7}$

Q 19 | Page 58

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

• 1

• −1

•  0

• None of these

Q 19 | Page 58

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

• 1

• −1

•  0

• None of these

Q 20 | Page 58

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

• 1

• 0

• −1

•  None of these

Q 20 | Page 58

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

• 1

• 0

• −1

•  None of these

Q 21 | Page 58

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

• −1

• 0

• 1

• None of these

Q 21 | Page 58

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

• −1

• 0

• 1

• None of these

Q 22 | Page 58

The value of

$\frac{\cos \left( 90°- \theta \right) \sec \left( 90°- \theta \right) \tan \theta}{cosec \left( 90°- \theta \right) \sin \left( 90° - \theta \right) \cot \left( 90°- \theta \right)} + \frac{\tan \left( 90° - \theta \right)}{\cot \theta}$

• 1

• − 1

•  2

•  −2

Q 22 | Page 58

The value of

$\frac{\cos \left( 90°- \theta \right) \sec \left( 90°- \theta \right) \tan \theta}{cosec \left( 90°- \theta \right) \sin \left( 90° - \theta \right) \cot \left( 90°- \theta \right)} + \frac{\tan \left( 90° - \theta \right)}{\cot \theta}$

• 1

• − 1

•  2

•  −2

Q 23 | Page 58

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

•  1

• −1

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

Q 23 | Page 58

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

•  1

• −1

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

Q 24 | Page 58

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

• cot2 A

• cot2 B

• −tan2 A

• −cot2 A

Q 24 | Page 58

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

• cot2 A

• cot2 B

• −tan2 A

• −cot2 A

Q 25 | Page 58

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

•  1

•  0

•  −1

• $1 + \sqrt{3}$

Q 25 | Page 58

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

•  1

•  0

•  −1

• $1 + \sqrt{3}$

Q 26 | Page 58

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

• sin 60°

• cos 60°

•  tan 60°

• sin 30°

Q 26 | Page 58

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

• sin 60°

• cos 60°

•  tan 60°

• sin 30°

Q 27 | Page 58

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

• tan 90°

• 1

• sin 45°

• sin 0°

Q 27 | Page 58

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

• tan 90°

• 1

• sin 45°

• sin 0°

Q 28 | Page 58

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

•  30°

• 45°

•  60°

Q 28 | Page 58

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

•  30°

• 45°

•  60°

Q 29 | Page 58

$\frac{2 \tan 30°}{1 - \tan^2 30°}$  is equal to

• cos 60°

• sin 60°

•  tan 60°

• sin 30°

Q 29 | Page 58

$\frac{2 \tan 30°}{1 - \tan^2 30°}$  is equal to

• cos 60°

• sin 60°

•  tan 60°

• sin 30°

Q 30 | Page 58

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

• $\sin \frac{A}{2}$

• $\cos \frac{A}{2}$

• $- \sin \frac{A}{2}$

• $- \cos \frac{A}{2}$

Q 30 | Page 58

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

• $\sin \frac{A}{2}$

• $\cos \frac{A}{2}$

• $- \sin \frac{A}{2}$

• $- \cos \frac{A}{2}$

Q 31 | Page 58

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

• 0

•  3

Q 31 | Page 58

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

• 0

•  3

Q 32 | Page 58

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

• 4/sqrt3

• 4sqrt3

• 1

• 4

Q 32 | Page 58

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

• 4/sqrt3

• 4sqrt3

• 1

• 4

Q 33 | Page 59

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

•  −2

•  2

• 1

• 0

Q 33 | Page 59

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

•  −2

•  2

• 1

• 0

Q 34 | Page 59

In the following figure  the value of cos ϕ is

• $\frac{5}{4}$

• $\frac{5}{3}$

• $\frac{3}{5}$

• $\frac{4}{5}$

Q 34 | Page 59

In the following figure  the value of cos ϕ is

• $\frac{5}{4}$

• $\frac{5}{3}$

• $\frac{3}{5}$

• $\frac{4}{5}$

Q 35 | Page 59

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

• $\frac{12}{5}$

• $\frac{5}{12}$

• $\frac{13}{12}$

• $\frac{12}{13}$

Q 35 | Page 59

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

• $\frac{12}{5}$

• $\frac{5}{12}$

• $\frac{13}{12}$

• $\frac{12}{13}$

## Chapter 10: Trigonometric Ratios

Ex. 10.10Ex. 10.20Ex. 10.30Others

## RD Sharma solutions for Class 10 Mathematics chapter 10 - Trigonometric Ratios

RD Sharma solutions for Class 10 Maths chapter 10 (Trigonometric Ratios) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 10 Mathematics chapter 10 Trigonometric Ratios are Introduction to Trigonometry, Introduction to Trigonometry Examples and Solutions, Trigonometric Ratios, Trigonometric Ratios of an Acute Angle of a Right-angled Triangle, Trigonometric Ratios of Some Specific Angles, Trigonometric Ratios of Complementary Angles, Trigonometric Identities, Proof of Existence, Relationships Between the Ratios, Trigonometry Ratio of Zero Degree and Negative Angles, Application of Trigonometry, Heights and Distances, Trigonometric Ratios of Complementary Angles, Trigonometric Identities, Trigonometric Ratios in Terms of Coordinates of Point, Angles in Standard Position.

Using RD Sharma Class 10 solutions Trigonometric Ratios exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 10 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 10 Trigonometric Ratios Class 10 extra questions for Maths and can use Shaalaa.com to keep it handy for your exam preparation

S