CBSE Class 10 Introduction to Trigonometry Sure Shot Questions Set 07

Question. If \( \sin \alpha = \frac{1}{\sqrt{2}} \) and \( \tan \beta = 1 \), find the value of \( \sin(\alpha + \beta) \), where \( \alpha \) and \( \beta \) both are acute.
Answer: 1

Question. If \( \cos \alpha = \frac{1}{2} \) and \( \tan \beta = \frac{1}{\sqrt{3}} \), find the value of \( \sin(\alpha + \beta) \), where \( \alpha \) and \( \beta \) both are acute.
Answer: 1

Question. Without using trigonometric tables evaluate the following :
(i) \( \frac{\sin 20^\circ}{\cos 70^\circ} \)
(ii) \( \frac{\cos 19^\circ}{\sin 71^\circ} \)
(iii) \( \frac{\sin 21^\circ}{\cos 69^\circ} \)
(iv) \( \frac{\tan 10^\circ}{\cot 80^\circ} \)
(v) \( \frac{\sec 11^\circ}{\csc 79^\circ} \)
(vi) \( \frac{\sin 20^\circ 30'}{\cos 69^\circ 30'} \)
Answer: (i) 1, (ii) 1, (iii) 1, (iv) 1, (v) 1, (vi) 1

Question. Without using trigonometric tables evaluate the following :
(i) \( \left( \frac{\sin 49^\circ}{\cos 41^\circ} \right)^2 + \left( \frac{\cos 41^\circ}{\sin 49^\circ} \right)^2 \)
(ii) \( \frac{\cot 40^\circ}{\tan 50^\circ} - \frac{1}{2} \left( \frac{\cot 35^\circ}{\sin 55^\circ} \right) \)
Answer: (i) 2, (ii) \( \frac{1}{2} \)

Question. Without using trigonometric tables evaluate the following :
(i) \( \frac{\tan 35^\circ}{\cot 55^\circ} + \frac{\cot 78^\circ}{\tan 12^\circ} - 1 \)
(ii) \( \csc^2 67^\circ - \tan^2 23^\circ \)
Answer: (i) 1, (ii) 1

Question. Without using trigonometric tables evaluate the following :
(i) \( \sin^2 20^\circ + \sin^2 70^\circ - \tan^2 45^\circ \)
(ii) \( \sec 50^\circ \sin 40^\circ + \cos 40^\circ \csc 50^\circ \)
Answer: (i) 0, (ii) 2

Question. Without using trigonometric tables prove the following :
(i) \( \tan 20^\circ \tan 35^\circ \tan 45^\circ \tan 55^\circ \tan 70^\circ = 1 \)
(ii) \( \sin 48^\circ \sec 42^\circ + \cos 48^\circ \csc 42^\circ = 2 \)
(iii) \( \sin 63^\circ \cos 27^\circ + \cos 63^\circ \sin 27^\circ = 1 \)
(iv) \( \frac{\sin 70^\circ}{\cos 20^\circ} + \frac{\csc 20^\circ}{\sec 70^\circ} - \cos 70^\circ \csc 20^\circ = 1 \)
(v) \( \frac{\cos 80^\circ}{\sin 10^\circ} + \cos 59^\circ \csc 31^\circ = 2 \)
Answer: Proved

Question. Without using trigonometric tables, evaluate each of the following :
(i) \( \sec^2 10^\circ - \cot^2 80^\circ + \frac{\sin 15^\circ \cos 75^\circ + \cos 15^\circ \sin 75^\circ}{\cos \theta \sin(90^\circ - \theta) + \sin \theta \cos(90^\circ - \theta)} \)
(ii) \( \sin(50^\circ + \theta) - \cos(40^\circ - \theta) + \tan 1^\circ \tan 10^\circ \tan 20^\circ \tan 70^\circ \tan 80^\circ \tan 89^\circ \)
(iii) \( \cot \theta \tan(90^\circ - \theta) - \sec(90^\circ - \theta) \csc \theta + \sin^2 25^\circ + \sin^2 65^\circ + \sqrt{3}(\tan 5^\circ \tan 45^\circ \tan 85^\circ) \)
(iv) \( \cot \theta \tan(90^\circ - \theta) - \sec(90^\circ - \theta) \csc \theta + \sqrt{3}(\tan 5^\circ \tan 30^\circ \tan 85^\circ) + \sin^2 25^\circ + \sin^2 65^\circ \)
(v) \( \frac{- \tan \theta \cot(90^\circ - \theta) + \sec \theta \csc(90^\circ - \theta)}{\tan 10^\circ \tan 20^\circ \tan 45^\circ \tan 70^\circ \tan 80^\circ} + \frac{\sin^2 35^\circ + \sin^2 55^\circ}{\tan 10^\circ \tan 20^\circ \tan 45^\circ \tan 70^\circ \tan 80^\circ} \)
Answer: (i) 2, (ii) 1, (iii) \( \sqrt{3} \), (iv) 1, (v) 2

Question. The round balloon of radius r subtends an angle \( \alpha \) at the eye of the observer while the angle of elevation of its centre is \( \beta \). Prove that the height of the centre of the balloon is r \( \sin \beta \csc \alpha/2 \).
Answer: Proved

Question. If \( \tan \theta = 8/15 \) and \( 0^\circ < \theta < 90^\circ \), find \( \sin \theta \).
Answer: 8/17

Question. If \( \sin \theta = 8/17 \) and \( 0^\circ < \theta < 90^\circ \), find \( \tan \theta \).
Answer: 8/15

Question. If \( \sin A = \frac{24}{25} \), find the value of \( \tan A + \sec A \), where \( 0^\circ < A < 90^\circ \).
Answer: 7

Question. If \( 5 \tan \theta = 12 \), find the value of \( \frac{2 \cos \theta + \sin \theta}{\sin \theta - \cos \theta} \).
Answer: 22/7

Question. If \( \tan \theta = \frac{3}{4} \), find the value of \( \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \).
Answer: 1/9

Question. If \( \tan \theta = \frac{12}{5} \), find the value of \( \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \).
Answer: 25

Question. If \( \tan A = \frac{1}{2} \) and \( \tan B = \frac{1}{3} \), by using \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \), prove that \( A + B = 45^\circ \).
Answer: Proved

Question. If \( 4 \tan \theta = 3 \), find the value of \( \frac{4 \sin \theta - 2 \cos \theta}{4 \sin \theta + 3 \cos \theta} \).
Answer: 1/6

Question. If \( \csc \theta = \frac{13}{12} \), find the value of \( \frac{2 \sin \theta - 3 \cos \theta}{4 \sin \theta - 9 \cos \theta} \).
Answer: 3

Question. Find the value of \( \frac{3\pi}{5} \) radians in degrees.
Answer: 108°

Question. Find the value of \( 150^\circ \) in radians.
Answer: \( \left( \frac{5\pi}{6} \right)^c \)

Question. If \( \sin \theta = \frac{5}{13} \), then find the values of \( \tan \theta \) and \( \sec \theta \).
Answer: 5/12 and 13/12

Question. If \( \tan \theta = \frac{x}{y} \), then find the value of \( \frac{x \sin \theta + y \cos \theta}{x \sin \theta - y \cos \theta} \).
Answer: \( \frac{x^2 + y^2}{x^2 - y^2} \)

Question. If \( 5 \tan \theta = 4 \), find the value of \( \frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 2 \cos \theta} \).
Answer: 1/6

Question. If \( 16 \cot x = 12 \), then find the value of \( \frac{\sin x - \cos x}{\sin x + \cos x} \).
Answer: 1/7

Question. If \( \tan \theta = (3/4) \) and \( 0 < \theta < 90^\circ \), then find the value of \( (\sin \theta \cos \theta) \).
Answer: 12/25

Question. If \( 8 \tan x = 15 \), then find the value of \( (\sin x - \cos x) \).
Answer: 7/17

Question. If \( \tan \theta = \frac{1}{\sqrt{7}} \), then find the value of \( \frac{\csc^2 \theta - \sec^2 \theta}{\csc^2 \theta + \sec^2 \theta} \).
Answer: 3/4

Question. If \( \cot \theta = \frac{1}{\sqrt{3}} \), then find the value of \( \frac{1 - \cos^2 \theta}{2 - \sin^2 \theta} \).
Answer: 3/5

Question. If \( \tan \theta = \frac{4}{3} \), then find the value of \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \).
Answer: 1/3

Question. If \( 3 \cos \theta = 5 \sin \theta \), then find the value of \( \frac{5 \sin \theta - 2 \sec^3 \theta + 2 \cos \theta}{5 \sin \theta + 2 \sec^3 \theta - 2 \cos \theta} \).
Answer: 271/979

Question. If \( \tan \theta = (3/4) \), then find the value of \( (\cos^2 \theta - \sin^2 \theta) \).
Answer: 7/25

Question. Find the value of \( \tan 75^\circ \).
Answer: \( 2 - \sqrt{3} \)

Question. If \( \tan \theta = \frac{a}{x} \), then find the value of \( \frac{x}{\sqrt{a^2 + x^2}} \).
Answer: \( \cos \theta \)

Question. If \( 3 \sin x + 5 \cos x = 5 \), then the value of \( (3 \cos x - 5 \sin x)^2 \).
Answer: 9

Question. Find the value of \( (\sin A + \cos A)^2 + (\sin A - \cos A)^2 \).
Answer: 2

Question. Find the value of \( \sqrt{\frac{1 + \sin A}{1 - \sin A}} \).
Answer: \( \sec A + \tan A \)

Question. Find the value of \( \sqrt{\frac{1 - \sin A}{1 + \sin A}} \).
Answer: \( \sec A - \tan A \)

Question. Find the value of \( \sqrt{\frac{1 - \cos x}{1 + \cos x}} \).
Answer: \( \csc x - \cot x \)

Question. Find the value of \( \sqrt{\frac{1 + \cos x}{1 - \cos x}} \).
Answer: \( \csc x + \cot x \)

Question. Find the value of \( \sqrt{\frac{\sec x - \tan x}{\sec x + \tan x}} \).
Answer: \( \sec x - \tan x \)

Question. Find the value of \( \frac{\cot \theta}{\cot \theta - \cot 3\theta} + \frac{\tan \theta}{\tan \theta - \tan 3\theta} \).
Answer: 1

Question. Find the value of \( \frac{\sin A + \sin B}{\cos A + \cos B} + \frac{\cos A - \cos B}{\sin A - \sin B} \).
Answer: 0

Question. Find the value of \( \sin 15^\circ \).
Answer: \( \frac{\sqrt{3} - 1}{2\sqrt{2}} \)

Question. Find the value of \( (\sin 40^\circ - \cos 50^\circ) \).
Answer: 0

Question. If \( x = r \sin A \cos B \), \( y = r \sin A \sin B \) and \( z = r \cos A \), then which is correct ?
(a) \( x^2 + y^2 + z^2 = r^2 \)
(b) \( x^2 - y^2 + z^2 = r^2 \)
(c) \( x^2 + y^2 - z^2 = r^2 \)
(d) \( -x^2 + y^2 + z^2 = r^2 \)
Answer: (a) \( x^2 + y^2 + z^2 = r^2 \)

Question. Find the value of \( (\cot 15^\circ \cot 16^\circ \cot 17^\circ \dots \cot 73^\circ \cot 74^\circ \cot 75^\circ) \).
Answer: 1