Question Bank — Introduction to Trigonometry

QUESTION BANK

CBSE Class 10 Mathematics – Chapter 8: Introduction to Trigonometry

Section A — MCQs (10 questions, 1 mark each)

1. In a right triangle ABC right-angled at B, if sin A = 3/5, then cos A is equal to
   (A) 3/5 (B) 4/5 (C) 5/4 (D) 5/3

2. The value of tan 45° + cot 45° is
   (A) 0 (B) 1 (C) 2 (D) Not defined

3. If cos A = 12/13, then sec A equals
   (A) 13/12 (B) 12/13 (C) 5/13 (D) 13/5

4. sin² 60° + cos² 60° equals
   (A) 0 (B) 1/2 (C) 1 (D) √3/2

5. In ΔABC right-angled at C, if tan A = 1, then ∠A equals
   (A) 30° (B) 45° (C) 60° (D) 90°

6. The value of cosec 30° – sec 60° is
   (A) 1 (B) 0 (C) 2 (D) –1

7. Assertion (A): In a right triangle, the value of sin A is always less than or equal to 1.
   Reason (R): Hypotenuse is the longest side of a right triangle.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false but R is true.

8. Assertion (A): tan 0° is defined and equals 0.
   Reason (R): sin 0° = 0 and cos 0° = 1.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false but R is true.

9. If cot θ = 8/15, then the value of tan θ is
   (A) 15/8 (B) 8/15 (C) 17/8 (D) 8/17

10. sin 30° cos 60° + cos 30° sin 60° equals
    (A) 1/2 (B) √3/2 (C) 1 (D) 0

Section B — Very Short Answer (6 questions, 2 marks each)

1. If tan A = 4/3, find the value of sin A and cos A.

2. Evaluate: 2 tan² 45° + cos² 30° – sin² 60°.

3. In ΔABC right-angled at B, AB = 24 cm and BC = 7 cm. Find sin A and cos C.

4. If sin A = 3/4, find cos A and tan A.

5. State the values of sin 0° and cos 90° as defined in the chapter.

6. If cot θ = 7/8, evaluate (1 + sin θ)(1 – sin θ) / [(1 + cos θ)(1 – cos θ)].

Section C — Short Answer (5 questions, 3 marks each)

1. Given sec θ = 13/12, calculate all other trigonometric ratios of θ.

2. In ΔPQR right-angled at Q, PQ = 3 cm and PR = 6 cm. Find ∠QPR and ∠PRQ.

3. If 3 cot A = 4, check whether (1 – tan² A)/(1 + tan² A) equals cos² A – sin² A.

4. In ΔABC right-angled at B, tan A = 1/3. Find the value of (i) sin A cos C + cos A sin C and (ii) cos A cos C – sin A sin C.

5. Prove that if ∠B and ∠Q are acute angles such that sin B = sin Q, then ∠B = ∠Q.

Section D — Long Answer (3 questions, 5 marks each)

1. In ΔACB right-angled at C, AB = 29 units, BC = 21 units and ∠ABC = θ. Determine the values of (i) cos² θ + sin² θ and (ii) cos² θ – sin² θ. (Multi-step problem)

2. Prove that cos² A + sin² A = 1 using the Pythagoras theorem in a right triangle. Hence prove the identity 1 + tan² A = sec² A for 0° ≤ A < 90°.

3. In ΔOPQ right-angled at P, OP = 7 cm and OQ – PQ = 1 cm. Determine sin Q and cos Q. Also verify that 2 sin Q cos Q = sin 2Q using the values obtained.

Section E — Case/Source-Based (2 questions, 4 marks each)

Case 1

A student is standing at a point on the ground looking at the top of Qutub Minar. A right triangle is imagined with the line of sight as hypotenuse. Let the distance of the student from the base be 50 m and the angle of elevation be 60°.

Sub-questions:
(a) Which trigonometric ratio will help find the height of the Minar?
(b) Write the ratio using the given angle.
(c) Calculate the height of the Minar.
(d) If the angle were 30° instead, what would be the new height?

Case 2

A girl is sitting on a balcony 12 m above the ground on one bank of a river. She looks at a flower pot on the opposite bank such that the angle of depression is 30°. A right triangle is formed by the line of sight.

Sub-questions:
(a) Which side of the triangle represents the width of the river?
(b) Which trigonometric ratio relates the height and the width?
(c) Find the width of the river.
(d) If the angle of depression increases to 60°, how does the width change?