NCERT Solutions
Class 10 - Mathematics

Chapter 8: INTRODUCTION TO TRIGONOMETRY

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

(i) sin A, cos A

(ii) sin C, cos C

In Fig. 8.13, find tan P − cot R.

If sin A = \(\dfrac{3}{4}\), calculate cos A and tan A.

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

Given sec θ = \(\dfrac{13}{12}\), 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 = \(\dfrac{7}{8}\), evaluate:

(i) \(\dfrac{(1 + \sin θ)(1 - \sin θ)}{(1 + \cos θ)(1 - \cos θ)}\)

(ii) cot² θ

If 3 cot A = 4, check whether \(\dfrac{1 - \tan^2 A}{1 + \tan^2 A} = \cos^2 A - \sin^2 A\) or not.

In triangle ABC, right-angled at B, if tan A = \(\dfrac{1}{\sqrt{3}}\), find:

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

(ii) cos A cos C − sin A sin C

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

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 = \(\dfrac{12}{5}\) 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 = \(\dfrac{4}{3}\) for some angle θ.

Evaluate the following:

1. \(\sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ\)
2. \(2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ\)
3. \(\dfrac{\cos 45^\circ}{\sec 30^\circ + \text{cosec } 30^\circ}\)
4. \(\dfrac{\sin 30^\circ + \tan 45^\circ - \text{cosec } 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ}\)
5. \(\dfrac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}\)

Choose the correct option and justify your choice:

1. \(\dfrac{2 \tan 30^\circ}{1 + \tan^2 30^\circ} =\)
   (A) \(\sin 60^\circ\)   (B) \(\cos 60^\circ\)   (C) \(\tan 60^\circ\)   (D) \(\sin 30^\circ\)
2. \(\dfrac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ} =\)
   (A) \(\tan 90^\circ\)   (B) 1   (C) \(\sin 45^\circ\)   (D) 0
3. \(\sin 2A = 2 \sin A\) is true when \(A =\)
   (A) \(0^\circ\)   (B) \(30^\circ\)   (C) \(45^\circ\)   (D) \(60^\circ\)
4. \(\dfrac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} =\)
   (A) \(\cos 60^\circ\)   (B) \(\sin 60^\circ\)   (C) \(\tan 60^\circ\)   (D) \(\sin 30^\circ\)

If \(\tan (A + B) = \sqrt{3}\) and \(\tan (A - B) = \dfrac{1}{\sqrt{3}}\); \(0^\circ < A + B \leq 90^\circ\), \(A > B\), find \(A\) and \(B\).

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

1. \(\sin (A + B) = \sin A + \sin B\).
2. The value of \(\sin \theta\) increases as \(\theta\) increases.
3. The value of \(\cos \theta\) increases as \(\theta\) increases.
4. \(\sin \theta = \cos \theta\) for all values of \(\theta\).
5. \(\cot A\) is not defined for \(A = 0^\circ\).

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

Write all the other trigonometric ratios of \(\angle A\) in terms of sec A.

Choose the correct option. Justify your choice:

1. \(9 \sec^2 A - 9 \tan^2 A =\)
   (A) 1   (B) 9   (C) 8   (D) 0
2. \((1 + \tan \theta + \sec \theta)(1 + \cot \theta - \text{cosec } \theta) =\)
   (A) 0   (B) 1   (C) 2   (D) -1
3. \((\sec A + \tan A)(1 - \sin A) =\)
   (A) \sec A   (B) \sin A   (C) \text{cosec } A   (D) \cos A
4. \(\dfrac{1 + \tan^2 A}{1 + \cot^2 A} =\)
   (A) \sec^2 A   (B) -1   (C) \cot^2 A   (D) \tan^2 A

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

1. \((\text{cosec } \theta - \cot \theta)^2 = \dfrac{1 - \cos \theta}{1 + \cos \theta}\)
2. \(\dfrac{\cos A}{1 + \sin A} + \dfrac{1 + \sin A}{\cos A} = 2 \sec A\)
3. \(\dfrac{\tan \theta}{1 - \cot \theta} + \dfrac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \text{cosec } \theta\)
4. \(\dfrac{1 + \sec A}{\sec A} = \dfrac{\sin^2 A}{1 - \cos A}\)
5. \(\dfrac{\cos A - \sin A + 1}{\cos A + \sin A - 1} = \text{cosec } A + \cot A\)
6. \(\sqrt{\dfrac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A\)
7. \(\dfrac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta\)
8. \((\sin A + \text{cosec} A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A\)
9. \((\text{cosec } A - \sin A)(\sec A - \cos A) = \dfrac{1}{\tan A + \cot A}\)
10. \(\dfrac{1 + \tan^2 A}{1 + \cot^2 A} = \left( \dfrac{1 - \tan A}{1 - \cot A} \right)^2 = \tan^2 A\)