Class 10 Mathematics – Chapter 8: Introduction to Trigonometry

Exercise 8.2 – NCERT Solutions

Q1. Evaluate the following.

(i) \(\sin60^\circ\cos30^\circ+\sin30^\circ\cos60^\circ\)

\(=\frac{\sqrt3}{2}\cdot\frac{\sqrt3}{2}+\frac12\cdot\frac12=\frac34+\frac14=1\).

Answer: 1

(ii) \(2\tan^245^\circ+\cos^230^\circ-\sin^260^\circ\)

\(=2(1)^2+\left(\frac{\sqrt3}{2}\right)^2-\left(\frac{\sqrt3}{2}\right)^2=2\).

Answer: 2

(iii) \(\frac{\cos45^\circ}{\sec30^\circ+\cosec30^\circ}\)

\(=\frac{1/\sqrt2}{2/\sqrt3+2}=\frac{3\sqrt2-\sqrt6}{8}\).

Answer: \(\frac{3\sqrt2-\sqrt6}{8}\)

(iv) \(\frac{\sin30^\circ+\tan45^\circ-\cosec60^\circ}{\sec30^\circ+\cos60^\circ+\cot45^\circ}\)

\(=\frac{\frac12+1-\frac2{\sqrt3}}{\frac2{\sqrt3}+\frac12+1}=\frac{43-24\sqrt3}{11}\).

Answer: \(\frac{43-24\sqrt3}{11}\)

(v) \(\frac{5\cos^260^\circ+4\sec^230^\circ-\tan^245^\circ}{\sin^230^\circ+\cos^230^\circ}\)

The denominator is 1. The numerator is \(5\cdot\frac14+4\cdot\frac43-1=\frac{67}{12}\).

Answer: \(\frac{67}{12}\)

Q2. Choose the correct option and justify your choice.

(i) \(\frac{2\tan^230^\circ}{1+\tan^230^\circ}=\frac{2/3}{4/3}=\frac12=\cos60^\circ\).

Correct option: (B) cos 60°

(ii) \(\frac{1-\tan^245^\circ}{1+\tan^245^\circ}=\frac{1-1}{1+1}=0\).

Correct option: (D) 0

(iii) \(\sin2A=2\sin A\) gives \(2\sin A\cos A=2\sin A\). Hence \(\sin A(\cos A-1)=0\). The listed value is \(A=0^\circ\).

Correct option: (A) 0°

(iv) \(\frac{2\tan30^\circ}{1-\tan^230^\circ}=\frac{2/\sqrt3}{2/3}=\sqrt3=\tan60^\circ\).

Correct option: (C) tan 60°

Q3. If tan(A + B) = 3 and tan(A − B) = 1/3; 0° < A + B ≤ 90°, A > B, find A and B.

Let \(A+B=\alpha\) and \(A-B=\beta\). Then \(\tan\alpha=3\) and \(\tan\beta=\frac13\).

Since \(\tan\alpha\tan\beta=1\), and both angles are acute, \(\alpha+\beta=90^\circ\). Therefore,

\(2A=(A+B)+(A-B)=90^\circ\), so \(A=45^\circ\).

Also, \(\tan B=\tan(A-(A-B))=\frac{1-1/3}{1+1/3}=\frac12\).

Hence, \(\boxed{A=45^\circ,\quad B=\tan^{-1}\left(\frac12\right)\approx26.565^\circ}\).

Q4. State whether the following are true or false. Justify your answer.

(i) \(\sin(A+B)=\sin A+\sin B\). False. The addition formula contains the products \(\sin A\cos B+\cos A\sin B\).

(ii) The value of sin θ increases as θ increases. True for \(0^\circ\le\theta\le90^\circ\), the range considered in this chapter.

(iii) The value of cos θ increases as θ increases. False. In the interval \(0^\circ\le\theta\le90^\circ\), cos θ decreases from 1 to 0.

(iv) \(\sin\theta=\cos\theta\) for all values of θ. False. They are equal only for particular angles, such as 45° in the first quadrant.

(v) cot A is not defined for A = 0°. True. \(\cot0^\circ=\frac{\cos0^\circ}{\sin0^\circ}=\frac10\), which is undefined.