NCERT solutions for Mathematics [English] Class 10 chapter 8 - Introduction to Trigonometry [Latest edition]

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 Given Figure, find tan P – cot R.

If sin A = `3/4`, calculate cos A and tan A.

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

Given sec θ = `13/12`, calculate all other trigonometric ratios.

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

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.

In ΔABC, right angled at B. If tan A = `1/sqrt3` , find the value of

1.  sin A cos C + cos A sin C
2. cos A cos C − sin A sin C

In ΔPQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.

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.

cot A is the product of cot and A.

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

sin θ = `4/3`, for some angle θ.

Evaluate the following in the simplest form:

sin 60° cos 30° + cos 60° sin 30°

Evaluate the following:

2tan² 45° + cos² 30° − sin² 60°

Evaluate the following:

`(cos 45°)/(sec 30° + cosec  30°)`

Evaluate the following:

`(sin 30° +  tan 45° –  cosec  60°)/(sec 30° +  cos 60° +  cot 45°)`

Evaluate the following:

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

`(2 tan 30°)/(1+tan^2 30°)` = ______.

`(1- tan^2 45°)/(1+tan^2 45°)` = ______

sin 2A = 2 sin A is true when A = ______.

`(2 tan 30°)/(1-tan^2 30°)` = ______.

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

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

sin (A + B) = sin A + sin B

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

The value of sinθ increases as θ increases.

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

The value of cos θ increases as θ increases.

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

sinθ = cosθ for all values of θ.

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

cot A is not defined for A = 0°.

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Write all the other trigonometric ratios of ∠A in terms of sec A.

9 sec² A − 9 tan² A = ______.

(1 + tan θ + sec θ) (1 + cot θ − cosec θ) = ______.

(secA + tanA) (1 − sinA) = ______.

`(1+tan^2A)/(1+cot^2A)` = ______.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(cosec  θ  – cot θ)^2 = (1-cos theta)/(1 + cos theta)`

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A`

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(tan theta)/(1-cot theta) + (cot theta)/(1-tan theta) = 1+secthetacosectheta`

[Hint: Write the expression in terms of sinθ and cosθ]

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(1+ secA)/sec A = (sin^2A)/(1-cosA)` 

[Hint : Simplify LHS and RHS separately.]

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA ` using the identity cosec² A = 1 cot² A.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`sqrt((1+sinA)/(1-sinA)) = secA + tanA`

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta`

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

(sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

(cosec A - sin A) (sec A - cos A) = `1/(tanA+cotA)` 

[Hint: Simplify LHS and RHS separately.] 

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`((1+tan^2A)/(1+cot^2A))=((1-tanA)/(1-cotA))^2=tan^2A`