8. Trigonometry

1. In  ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine :

(i) sin A, cos A

(ii) sin C, cos C

2. In Fig. 8.13, find tan P – cot R.

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

4. Given 15 cot A = 8, find sin A and sec A.

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

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

7. If cot θ = 7/8 , evaluate :

(i) (1 + sin θ)(1 - sin θ)/(1 + cos θ)(1 - cos )

(ii) cot² θ

8. If 3 cot A = 4, check whether 1 - tan² A/1 + tan ² A = cos² A – sin²A or not.

9. In triangle ABC, right-angled at B, if tan A = 1/√3, find the value of:

(i) sin A cos C + cos A sin C

(ii) cos A cos C – sin A sin C

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

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

(i) The value of tan A is always less than 1.

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

(iii) cos A is the abbreviation used for the cosecant of angle A.

(iv) cot A is the product of cot and A.

(v) sin θ = 4/3 for some angle θ.

1. Evaluate the following :

(i) sin 60° cos 30° + sin 30° cos 60°

(ii) 2 tan2 45° + cos2 30° – sin2 60°

(iii) cos 45°/sec 30° + cosec 30°

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

(v) 5 cos² 60° + 4 sec² 30° tan² 45°/sin² 30° + cos² 30°

2. Choose the correct option and justify your choice :

(i) 2 tan 30° 1 tan 30°

(A) sin 60° (B) cos 60° (C) tan 60° (D) sin 30°

(ii) 1 - tan 45° / 1 + tan 45°

(A) tan 90° (B) 1 (C) sin 45° (D) 0

(iii) sin 2A = 2 sin A is true when A =

(A) 0° (B) 30° (C) 45° (D) 60°

(iv) 2 tan 30°/1 - tan² 30° =

(A) cos 60° (B) sin 60° (C) tan 60° (D) sin 30°

3. If tan (A + B) = √3 and tan (A – B) = 1/√3; 0° < A + B ≤ 90°; A > B, find A and B.

(i) 1/√2    (ii) 7√5    (iii) 6 + √2

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

(i) sin (A + B) = sin A + sin B.

(ii) The value of sin θ increases as θ increases.

(iii) The value of cos θ increases as θ increases.

(iv) sin θ = cos θ for all values of θ.

(v) cot A is not defined for A = 0°.

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

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

3. Choose the correct option. Justify your choice.

(i) 9 sec² A – 9 tan² A =

(A) 1   (B) 9   (C) 8   (D) 0

(ii) (1 + tan θ + sec θ) (1 + cot θ – cosec θ) =

(A) 0   (B) 1   (C) 2   (D) –1

(iii) (sec A + tan A) (1 – sin A) =

(A) sec A   (B) sin A   (C) cosec A   (D) cos A

(iv) 1 + tan² A/1 + cot² A =

(A) sec2 A   (B) –1   (C) cot2 A   (D) tan2 A

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

(i) (cosec θ – cot θ)2 = 1 - cos/1 + cos

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

(iii) tan θ/1 - cot θ/1- tan θ = 1 + sec θ cosec θ

[Hint : Write the expression in terms of sin θ and cos θ]

(iv) 1 + sec/sec A = sin² A/1 – cos A

[Hint : Simplify LHS and RHS separately]

(v) cos A – sin A + 1/cos A + sin A – 1 = cosec A + cot A, using the identity cosec² A = 1 + cot² A.

(vi) 1 + sin A/1 – sin A = sec A + tan A

(vii) sin θ - 2 sin³ θ/2 cos³θ = tan θ

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

(ix) (cosec A – sin A) (sec A – cos A) = 1/tan A + cot A

[

Hint : Simplify LHS and RHS separately]

(1 + tan² A/1 + cot² A) = (1 - tan A/1 - cot A) = tan² A