CBSE Class 10 Mathematics Introduction to Trigonometry Worksheet Set 03

Question. The value of \( (\sin 30^\circ + \cos 60^\circ) \)
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (a) 1

Question. The value of \( \frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ} \) is
(a) \( \frac{32}{35} \)
(b) \( \frac{14}{55} \)
(c) \( \frac{67}{12} \)
(d) \( \frac{19}{33} \)
Answer: (c) \( \frac{67}{12} \)

Question. The value of \( \frac{3 \sin 30^\circ + 4 \cos^2 45^\circ - \cot^2 30^\circ}{\cos^2 30^\circ + \sin^2 30^\circ} \) is
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{2}{5} \)
(d) \( \frac{3}{8} \)
Answer: (a) \( \frac{1}{2} \)

Question. Given \( \sin (A - B) = \frac{\sqrt{3}}{2} \) and \( \cos (A + B) = \frac{\sqrt{3}}{2} \). Then A and B respectively are
(a) 30°, 45°
(b) 45°, –15°
(c) 60°, 45°
(d) None of the options
Answer: (b) 45°, –15°

Question. The value of \( \sin 60^\circ \cdot \cos 30^\circ + \sin 30^\circ \cdot \cos 60^\circ \) is
(a) 0
(b) 1
(c) 2
(d) 8
Answer: (b) 1

Question. Value of \( \cos 0^\circ \cdot \cos 30^\circ \cdot \cos 45^\circ \cdot \cos 60^\circ \cdot \cos 90^\circ \) is
(a) 0
(b) 1
(c) 2
(d) 9
Answer: (a) 0

Question. The value of \( \left( \sin^2 \theta + \frac{1}{1 + \tan^2 \theta} \right) = \)
(a) 0
(b) 1
(c) 2
(d) 5
Answer: (b) 1

Question. The value of \( (1 + \tan^2 \theta)(1 - \sin \theta)(1 + \sin \theta) = \)
(a) 0
(b) 1
(c) 2
(d) None of the options
Answer: (b) 1

Question. If \( \tan(A + B) = \sqrt{3} \) and \( \tan(A - B) = \frac{1}{\sqrt{3}} \), A > B, then the value of A is
(a) A = 30°
(b) A = 60°
(c) A = 90°
(d) A = 45°
Answer: (d) A = 45°

Question. The value of \( \sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ \) is
(a) 1
(b) 2
(c) 11
(d) 0
Answer: (a) 1

Question. \( 2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\circ \) equals
(a) 1
(b) 2
(c) 5
(d) 6
Answer: (b) 2

Question. If \( \sin \theta = x \) and \( \sec \theta = y \), then the value of \( \cot \theta \) is
(a) \( xy \)
(b) \( 2xy \)
(c) \( \frac{1}{xy} \)
(d) \( x + y \)
Answer: (c) \( \frac{1}{xy} \)

Question. If \( (1 + \cos A)(1 - \cos A) = \frac{3}{4} \), the value of \( \sec A \) is
(a) 2
(b) –2
(c) ±2
(d) 0
Answer: (c) ±2

Question. If \( 15 \cot A = 8 \), then the value of \( \text{cosec } A \) is
(a) \( \frac{15}{12} \)
(b) \( \frac{13}{15} \)
(c) \( \frac{4}{15} \)
(d) \( \frac{17}{15} \)
Answer: (d) \( \frac{17}{15} \)

Question. Evaluate: \( 4 \sin^2 60^\circ + 3 \tan^2 30^\circ - 8 \sin 45^\circ \cos 45^\circ \)
(a) 0
(b) 1
(c) 2
(d) 5
Answer: (a) 0

Question. Evaluate: \( \frac{\sin 30^\circ + \tan 45^\circ - \text{cosec } 60^\circ}{\sec 30^\circ + \cos 60^\circ + \cot 45^\circ} \)
(a) \( \frac{3\sqrt{3} + 2}{3\sqrt{3} - 2} \)
(b) \( \frac{3\sqrt{3} - 4}{3\sqrt{3} + 4} \)
(c) \( \frac{3\sqrt{3} + 8}{3\sqrt{3} - 9} \)
(d) None of the options
Answer: (b) \( \frac{3\sqrt{3} - 4}{3\sqrt{3} + 4} \)

Question. The value of \( \frac{\cos 30^\circ + \sin 60^\circ}{1 + \cos 60^\circ + \sin 30^\circ} \) is
(a) \( \frac{\sqrt{3}}{2} \)
(b) \( \frac{2}{\sqrt{3}} \)
(c) \( \frac{1}{\sqrt{2}} \)
(d) 0
Answer: (a) \( \frac{\sqrt{3}}{2} \)

Question. ABC is a triangle right angled at C and \( AC = \sqrt{3} BC \). Then \( \angle ABC = \)
(a) 30°
(b) 60°
(c) 90°
(d) 0°
Answer: (b) 60°

Question. Assertion (A): In a right-angled triangle, if \( \tan \theta = \frac{3}{4} \), the greatest side of the triangle is 5 units.
Reason (R): \( (\text{Greatest side})^2 = (\text{Hypotenuse})^2 = (\text{Perpendicular})^2 + (\text{Base})^2 \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion (A): In a right-angled triangle, if \( \cos \theta = \frac{1}{2} \) and \( \sin \theta = \frac{\sqrt{3}}{2} \), then \( \tan \theta = \sqrt{3} \).
Reason (R): \( \frac{\sin \theta}{\cos \theta} \)
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Trigonometric Identities

An equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved.

Some trigonometric ratios are listed below:
(i) \( \sin^2 A + \cos^2 A = 1, 0^\circ \le A \le 90^\circ \)
(ii) \( 1 + \tan^2 A = \sec^2 A, 0^\circ \le A < 90^\circ \)
(iii) \( \cot^2 A + 1 = \text{cosec}^2 A, 0^\circ < A \le 90^\circ \)

Question. The value of \( \sin^6 \theta + \cos^6 \theta + 3\sin^2 \theta \cos^2 \theta \) is
(a) 0
(b) 1
(c) 2
(d) \( \frac{1}{4} \)
Answer: (b) 1

Question. The value of \( (\sin^4 \theta - \cos^4 \theta + 1) \text{cosec}^2 \theta \) is
(a) 0
(b) 1
(c) 2
(d) 5
Answer: (c) 2

Question. If \( \tan A = n \tan B \) and \( \sin A = m \sin B \), then \( \cos^2 A = \)
(a) \( \frac{m^2 - 1}{n^2 - 1} \)
(b) \( \frac{m^2 + 1}{n^2 + 1} \)
(c) \( \frac{1 - m^2}{1 + m^2} \)
(d) \( \frac{1 + m^2}{1 - m^2} \)
Answer: (a) \( \frac{m^2 - 1}{n^2 - 1} \)

Question. If \( 4 \tan \theta = 3 \), then \( \left[ \frac{4 \sin \theta - \cos \theta}{4 \sin \theta + \cos \theta} \right] \) is equal to
(a) \( \frac{2}{3} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{3}{4} \)
Answer: (c) \( \frac{1}{2} \)

Question. If \( \sin \theta - \cos \theta = 0 \), then the value of \( (\sin^4 \theta + \cos^4 \theta) \) is
(a) 1
(b) \( \frac{3}{4} \)
(c) \( \frac{1}{2} \)
(d) \( \frac{1}{4} \)
Answer: (c) \( \frac{1}{2} \)

Question. If \( 2 \sin \theta = \sqrt{3} \), then \( \theta = \)
(a) 30°
(b) 60°
(c) 45°
(d) 90°
Answer: (b) 60°

Question. The value of \( (1 + \tan^2 \theta)(1 - \sin \theta)(1 + \sin \theta) \) is
(a) 0
(b) 1
(c) 8
(d) 17
Answer: (b) 1

Question. The value of \( \cot^2 \theta - \frac{1}{\sin^2 \theta} \) is
(a) 0
(b) –1
(c) 2
(d) –8
Answer: (b) –1

Question. If \( \text{cosec } \theta + \cot \theta = x \), the value of \( \text{cosec } \theta - \cot \theta \) is
(a) \( x \)
(b) \( 2x \)
(c) \( \frac{x}{2} \)
(d) \( \frac{1}{x} \)
Answer: (d) \( \frac{1}{x} \)

Question. The magnitude of \( \theta \) in the equation \( \frac{\cos^2 \theta}{\cot^2 \theta - \cos^2 \theta} = 3 \) is
(a) 0°
(b) 30°
(c) 60°
(d) 90°
Answer: (c) 60°

Question. The value of \( (1 + \cot \theta - \text{cosec } \theta) (1 + \tan \theta + \sec \theta) \) is equal to
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. If \( 7 \sin^2 A + 3 \cos^2 A = 4 \), then \( \tan A = \)
(a) \( \frac{1}{2} \)
(b) \( \frac{1}{3} \)
(c) \( \frac{1}{\sqrt{2}} \)
(d) \( \frac{1}{\sqrt{3}} \)
Answer: (d) \( \frac{1}{\sqrt{3}} \)

Question. Assertion (A): \( \sin^2 67^\circ + \cos^2 67^\circ = 1 \).
Reason (R): For any value of \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion (A): The value of \( \sec^2 10^\circ - \cot^2 80^\circ \) is 1.
Reason (R): The value of \( \sin 30^\circ = \frac{1}{2} \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

Children were playing a game by making some right angled triangles on the plane sheet of paper. They assumed angles and their corresponding sides. They wanted to find all trigonometric ratio so they took a right angled triangle with two of its sides AC = 13 cm, BC = 12 cm and \( \angle ABC = 90^\circ \). They are unable to find trigonometric ratio. So help them to do so.

Question. Using the above data, the value of \( \sin A \) is
(a) \( \frac{12}{13} \)
(b) \( \frac{13}{12} \)
(c) \( \frac{5}{13} \)
(d) \( \frac{13}{5} \)
Answer: (a) \( \frac{12}{13} \)

Question. Using the above data, the value of \( \sin C \) is
(a) \( \frac{12}{13} \)
(b) \( \frac{13}{12} \)
(c) \( \frac{5}{13} \)
(d) \( \frac{13}{5} \)
Answer: (c) \( \frac{5}{13} \)