R.D. Sharma solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० chapter 10 - Trigonometric Ratios [Latest edition]

In the following, trigonometric ratios is given. Find the value of the other trigonometric ratios.

`sin A = 2/3`

In the following, trigonometric ratios is given. Find the value of the other trigonometric ratios.

`cos A = 4/5`

In the following, trigonometric ratios is given. Find the value of the other trigonometric ratios.

tan θ = 11

In the following, trigonometric ratios is given. Find the value of the other trigonometric ratios.

`cot theta = 12/5`

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

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

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

If `tan θ = a/b` then 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 `(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ) = 3`.

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

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

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

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

In the following figure, find tan P and cot R. Is tan P = cot R?

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

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.

If sin θ = `12/13`, find the value of `(sin^2 θ - cos^2 θ)/(2sin θ cos θ) × 1/(tan^2 θ)`.

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

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

If sin A = y, then express cos A and tan A is terms of y.

Evaluate the following:

sin 45° sin 30° + cos 45° cos 30°

Evaluate the following:

cos 60° cos 45° – sin 60° sin 45°

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:

`(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:

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

Evaluate the following:

`(5cos^2 60° +  4sec^2 30° - tan^2 45°)/(sin^2 30° +  cos^2 30°)`

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:

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

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)`

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 A tan B)`

Prove the following:

`(sqrt(3) + 1) (3 - cot 30^circ)` = tan³ 60° – 2 sin 60°

If `tan (A - B) = 1/sqrt(3)` and `tan (A + B) = sqrt(3)`, 0° < (A + B) ≤ 90° and A > B then find the values of A and B.

If tan (A + B) = 1 and tan(A-B)`=1/sqrt3` , 0° < A + B < 90°, A > B, then find the values of 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

If `4 cot^2 45^circ - sec^2 60^circ + sin^2 60^circ + p = 3/4`, find the value of p.

If A and B are acute angles such that sin (A – B) = 0 and 2 cos (A + B) – 1 = 0, then find angles A and B.

Find acute angles A and B, if `sin (A + 2B) = sqrt(3)/2` and cos(A + 4B) = 0, A > B.

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

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 a right triangle ABC, right angled at C, if ∠B = 60° and AB = 15units, find the remaining angles and sides.

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

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)`, find A + B.

If `tan A = sqrt(3)`, where A is an acute angle, then find the value of `(sin^2 A)/(1 + cos^2 A)`.

Evaluate the following:

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

Evaluate the following:

`cos 19^@/sin 71^@`

Evaluate the following:

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

Evaluate the following:

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

Evaluate the following:

`(cot 40^circ)/(tan 50^circ) - 1/2(cos 35^circ/sin 55^circ)`

Evaluate the following:

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

Evaluate the following:

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

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

Evaluate the following 

sec 50º sin 40° + cos 40º cosec 50º 

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

cos 78° + sec 78°

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

cosec 54° + sin 72°

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

cot 85° + cos 75°

Express 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}`

A, B and C are interior angles of a triangle ABC. Show that

If ∠A = 90°, then find the value of tan`(("B+C")/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`

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

Evaluate:

`((3tan41^circ)/(cot49^circ))^2 - ((sin35^circ sec55^circ)/(tan10^circ tan20^circ tan 60^circ tan 70^circ tan 80^circ))^2`

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

If A, B, C are the interior angles of a triangle ABC, prove that `\tan \frac{B+C}{2}=\cot \frac{A}{2}`

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

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.

If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.

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

Evaluate: `cos 58^@/sin 32^@  + sin 22^@/cos 68^@  - (cos 38^@ cosec 52^@)/(tan 18^@ tan 35^@ tan 60^@ tan 72^@ tan 55^@)`

If 2θ + 45° and 30° – θ are acute angles, find the degree measure of θ satisfying sin (2θ + 45°) = cos (30° – θ)

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 \[\frac{1}{cosec \theta}\]?

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 `tan θ = 1/sqrt(5)`, what is the value of `(cosec^2θ - sec^2θ)/(cosec^2θ + sec^2θ)`?

If `cot theta = 1/ sqrt(3)`, write the value of `((1 - cos^2 theta))/((2 - sin^2 theta))`.

Write the acute angle θ satisfying `sqrt(3) sin θ = cos θ`.

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

Evaluate: sin² 60° + 2tan 45° – cos² 30°.

If `sin A = 3/4`, calculate sec A.

If `tan A = 5/12`, find the value of (sin A + cos A) sec A.

In Figure 1, PS = 3 cm, QS = 4 cm, ∠PRQ = θ, ∠PSQ = 90°, PQ ⊥ RQ and RQ = 9 cm. Evaluate tan θ.

Evaluate:

2 sin² 30° sec 60° + tan² 60°.

If 2 sin (A + B) = `sqrt3` and cos (A − B) = 1, then find the measures of angles A and B. 0 ≤ A, B, (A + B) ≤ 90°.

Evaluate:

`2 sqrt2  cos 45^circ  sin 30^circ + 2 sqrt3  cos 30^circ`

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

Evaluate: `(5cos^2 60° +  4sec^2 30° - tan^2 45°)/(sin^2 30° + sin^2 60°)`

If `sin (A - B) = 1/2, cos (A + B) = 1/2; 0 < A + B ≤ 90^circ`. A > B; find ∠A and ∠B.

Evaluate:

`(2 tan 30^circ sec 60^circ tan 45^circ)/(1 - sin^2 60^circ)`

Evaluate:

`(5 tan 60^circ)/((sin^2 60^circ + cos^2 60^circ) tan 30^circ)`

If `tan A = sqrt(3)`, then find the value of cos² A – sin² А.

If x sin 60° + cos 30° – tan 45° = `sqrt(3)/2`, find the value of x.

If 4k = tan² 60° − 2 cosec² 30° − 2 tan² 30°, then find the value of k.

The value of (sin 30° + cos 30°) – (sin 60° + cos 60°) is ______.

The value of `(tan 30^circ)/(cot 60^circ)` is ______.

The value of (sin 45° + cos 45°)³ is ______.

The value of (sin 30° + cos 30°)² – (sin 60° – cos 60)² is ______.

The value of (cos² 23° – sin² 67°) is ______.

The value of the expression `{(sin^2 22^circ + sin^2 68^circ)/(cos^2 22^circ + cos^2 68^circ) + sin^2 63^circ + cos 63^circ sin 27^circ}` is ______.

Given that `sin α = 1/2 cos β = 1/2`, then α + β = ______.

Given that `sin (α - β) = 1/2` and `cos (α + β) = 1/2`, then α = ______ β = ______.

If 4 tan θ = 3, then is equal to `(4 sin - cos θ)/(4 sin θ + cos θ)` is equal to ______.

If cos 5θ = sin θ and 5θ < 90°, then the value of tan 3θ is ______.

The value of sec² 60° – tan² 60° is ______.

If A + B = 90°, then the value of tan² A – cot² B is ______.

The value of cos 1° cos 2° cos 3°.... cos 120° is ______.

If tan θ + cot θ = 2 and 0° < θ < 90° then tan¹⁰ θ + cot¹⁰ θ is equal to ______.

If sin θ + cos θ = 1 and 0° ≤ θ ≤ 90°, then the possible values of θ are ______.

If `sin (A + B) = cos(A - B) = sqrt(3)/2`, then cot 2A = ______.

If ΔABC is an isosceles right triangle right angled at B, then `(tan A + cot C)/(cot A + cot C)` = ______.

If α + β = 90° and `α = β/2`, then tan α tan β = ______.

If in a triangle ABC, angles A and B are complementary, then the value of cot C is ______.

The value of tan 25° tan 10° tan 80° tan 65° is ______.