# Introduction to Trigonometry NCERT Solutions For Class 10 Maths

Exercise 8.1 Questions

In Δ 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+sinθ ) (1 -sin θ ) /(1+ cosθ )(1- cosθ )
(ii) cot2θ

8. If 3 cot A = 4, check whether 1-tan2A/ 1 + tan2 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 Δ 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.      ### EXERCISE 8.2 Trigonometry Questions

1. Evaluate the following :
(i) sin 60° cos 30° + sin 30° cos 60° (ii) 2 tan2 45° + cos2 30° – sin260°
(iii) cos 45°/ sec 30° + cosec 30°
(iv) sin 30° + tan 45° – cosec 60°/sec 30° + cos 60+ cot 45°
(v) 5 cos260° + 4 sec230°- tan245°/sin230°/cos230°

2. Choose the correct option and justify your choice :
(i) 2 tan30°/1 +tan230°
(A) sin 60° (B) cos 60° (C) tan 60° (D) sin 30°

(ii) 1-tan245°/1 + tan245°
(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-tan230°
(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.

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°.    ### EXERCISE 8.3 Introduction to trigonometry Questions

1. Evaluate :
(i) sin 18°/cos 72°
(ii) tan 26°/cot 64°
(iii) cos 48° – sin 42° (iv) cosec 31° – sec 59°
2. Show that :
(i) tan 48° tan 23° tan 42° tan 67° = 1
(ii) cos 38° cos 52° – sin 38° sin 52° = 0
3. If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.
4. If tan A = cot B, prove that A + B = 90°.
5. If sec 4A = cosec (A – 20°), where 4A is an acute angle, find the value of A.
6. If A, B and C are interior angles of a triangle ABC, then show that
sin( B + C/2)=  cos A/2
7. Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

### EXERCISE 8.4 Introduction to trigonometry Questions

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. Evaluate :
(i) Sin263° sin227°/ Cos2 17° cos273°
(ii) sin 25° cos 65° + cos 25° sin 65°
4. 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) secA (B) sinA (C) cosec A (D) cos A
(iv) 1+ tan2A/ 1 + cot2A
(A) sec2 A (B) –1 (C) cot2 A (D) tan2 A
5. Prove the following identities, where the angles involved are acute angles for which the
expressions are defined.(cosec θ – cot θ)2 = 1 – cosθ /1 +cosθ
(ii) cos A /1 +sin A + 1+Sin A /Cos A = 2 sec A      