Chapter 8 - Introduction to Trigonometry

1. If \( \sin\theta=\frac{a}{b} \), then \( \cos\theta \) is equal to:

(A) \( \frac{b}{\sqrt{b^2-a^2}} \)

(B) \( \frac{b}{a} \)

(C) \( \frac{\sqrt{b^2-a^2}}{b} \)

(D) \( \frac{a}{\sqrt{b^2-a^2}} \)

[STD: 30/1/1 Q9, 30/1/3 Q7]

2. If \( \cos A=\frac12 \), then the value of \( \sin^2A+2\cos^2A \) is:

(A) \( \frac32 \)

(B) \( \frac54 \)

(C) \( 1 \)

(D) \( \frac12 \)

[STD: 30/1/1 Q10, 30/1/2 Q11, 30/1/3 Q8]

3. If \( 2\sin A=1 \), then the value of \( \tan A+\cot A \) is:

(A) \( \sqrt3 \)

(B) \( \frac4{\sqrt3} \)

(C) \( \frac{\sqrt3}{2} \)

(D) \( 1 \)

[STD: 30/2/1 Q11, 30/2/2 Q14, 30/2/3 Q5]

4. If \( \sin2\alpha=\frac{\sqrt3}{2} \), the value of \( \sin3\alpha \) is:

(A) \( \frac{3\sqrt3}{4} \)

(B) \( \frac12 \)

(C) \( 1 \)

(D) \( \frac{\sqrt3}{4} \)

[STD: 30/3/1 Q15, 30/3/2 Q1, 30/3/3 Q12]

5. Which of the following statements is false?

(A) \( \tan45^\circ=\cot45^\circ \)

(B) \( \sin90^\circ=\tan45^\circ \)

(C) \( \sin30^\circ=\cos30^\circ \)

(D) \( \sin45^\circ=\cos45^\circ \)

[BASIC: 430/1/1, 430/1/2, 430/1/3]

6. Find the value of \( \theta \) for which \( \sin2\theta=\tan45^\circ \).

(A) \(22.5^\circ\)

(B) \(30^\circ\)

(C) \(45^\circ\)

(D) \(90^\circ\)

[BASIC: 430/3/3]

7. The value of \( \frac{2\tan60^\circ}{1-\tan^260^\circ} \) is:

(A) \(-3\)

(B) \( \sqrt3 \)

(C) \(-\frac1{\sqrt3}\)

(D) \(-\sqrt3\)

[BASIC: 430/2/1, 430/2/2, 430/2/3]

8. The value of \( \frac13\cot^230^\circ-\frac12\sec^260^\circ \) is:

(A) \(-1\)

(B) \(-2\)

(C) \( \frac58 \)

(D) \( \frac78 \)

[STD: 30/3/2 Q11]

9. The value of \( \frac{1+\tan^2A}{1+\cot^2A} \) is:

(A) \( \tan^2A \)

(B) \(-1\)

(C) \(-\tan^2A\)

(D) \( \cot^2A \)

[STD: 30/4/1 Q12, 30/4/2 Q4, 30/4/3 Q15]

10. The value of \( \frac{\sec^2A-1}{\sin^2A} \) is same as:

(A) \( \cos^2A \)

(B) \( \sec^2A \)

(C) \(-\sec^2A\)

(D) \( \cot^2A \)

[STD: 30/4/3 Q9]

11. The simplest form of \( \frac{\sec A}{\sqrt{\sec^2A-1}} \) is:

(A) \( \sin A \)

(B) \( \tan A \)

(C) \( \csc A \)

(D) \( \cos A \)

[STD: 30/5/1 Q18, 30/5/2 Q9, 30/5/3 Q6]

12. The value of \( \tan^2A-\frac1{\cos^2A} \) is:

(A) More than 1

(B) 1

(C) 0

(D) \(-1\)

[BASIC: 430/1/1]

13. The value of \( \cot^2A-\frac1{\sin^2A} \) is:

(A) More than 1

(B) 1

(C) 0

(D) \(-1\)

[BASIC: 430/1/2]

14. The value of \( \frac1{\sec^2A}+\frac1{\csc^2A} \) is:

(A) More than 1

(B) 1

(C) 0

(D) \(-1\)

[BASIC: 430/1/3]

15. The value of \( (\sin^2A+\cos^2A)+(\sec^2A-\tan^2A)-(\cot^2A-\csc^2A) \) is:

(A) 1

(B) \(-1\)

(C) 3

(D) 2

[BASIC: 430/2/1]

16. The value of \( (\sin\theta+\cos\theta)^2+(\sin\theta-\cos\theta)^2 \) is:

(A) 1

(B) 2

(C) \(2+2\sin\theta\cos\theta\)

(D) \(2+4\sin\theta\cos\theta\)

[BASIC: 430/2/2]

17. The value of \( (\sec\theta-\cos\theta)^2+\sin^2\theta-\tan^2\theta \) is:

(A) 0

(B) 1

(C) 2

(D) 4

[BASIC: 430/2/3]

18. The value of \( \frac{\sec^230^\circ+\tan^230^\circ} {\sin^245^\circ+\cos^245^\circ} \) is:

(A) 1

(B) \( \frac53 \)

(C) \( \frac{13}{3} \)

(D) 7

[BASIC: 430/3/1]

19. The value of \( \frac{\cot^2A-\csc^2A}{\sin30^\circ+\cos60^\circ} \) is:

(A) 1

(B) \(-1\)

(C) \( \frac2{1+\sqrt3} \)

(D) \( -\frac2{1+\sqrt3} \)

[BASIC: 430/3/2]

20. Assertion (A): For \( \sin\theta=1 \), \( \cos\theta \) must be 0.

Reason (R): \( \sin^2\theta-\cos^2\theta=1 \).

[BASIC: 430/3/1 Q19]

21. Assertion (A): For an acute angle \( \theta \), \( \cot\theta=1\implies\csc\theta=2 \).

Reason (R): \( \csc^2\theta-\cot^2\theta=1 \).

[BASIC: 430/3/2 Q20]

22. Assertion (A): For an angle \( \theta \), \( \sec\theta=1\implies\tan\theta=0 \).

Reason (R): \( \sec^2\theta+\tan^2\theta=1 \).

[BASIC: 430/3/3 Q19]

23. Assertion (A): \( \tan2\theta \) is not defined at \( \theta=45^\circ \).

Reason (R): \( \sin90^\circ\ne\cos90^\circ \).

[STD: 30/3/1 Q20, 30/3/2 Q20]

24. Find the values of \(A\) and \(B\), where \(0^\circ\le A<90^\circ,\ 0^\circ\le B<90^\circ\), if \( \tan(A+B)=1 \) and \( \tan(A-B)=\frac1{\sqrt3} \).

[BASIC: 430/1/1, 430/1/2, 430/1/3]

25. If \( \sec(2A-B)=2 \) and \( \csc(A+B)=2 \), find the values of \(A\) and \(B\).

[STD: 30/5/2 Q22(a)]

26. Prove geometrically that \( \tan45^\circ=1 \).

[BASIC: 430/1/1, 430/1/2, 430/1/3]

27. If \( \sin3A=1 \), find the value of \( \cos2A-\tan^245^\circ \).

[BASIC: 430/3/1, 430/3/2, 430/3/3]

28. Evaluate: \( \frac{\sin^245^\circ}{\csc^230^\circ-\tan^245^\circ} \).

[BASIC: 430/2/3]

29. Evaluate: \( \frac{3\cos^230^\circ-6\csc^230^\circ}{\tan^260^\circ} \).

[STD: 30/4/1 Q23(b), 30/4/2 Q25(b)]

30. Evaluate: \( \sin^360^\circ-\frac{\tan30^\circ}{\cos^245^\circ} \).

[STD: 30/5/1 Q24(a), 30/5/3 Q22(a)]

31. Prove that \( \sqrt{\frac{1+\sin A}{1-\sin A}}=\sec A+\tan A \).

[STD: 30/4/1 Q23(a), 30/4/2 Q25(a)]

32. Prove that \( \frac{\tan\theta}{1+\tan^2\theta} +\frac{\cot\theta}{1+\cot^2\theta} =2\sin\theta\cos\theta \).

[STD: 30/4/3 Q21(a)]

33. Prove that \( 1+\frac{\cot^2\alpha}{1+\csc\alpha}=\csc\alpha \).

[STD: 30/2/2 Q22(b)]

34. If \( (\sec A+\tan A)(1-\sin A)=k\cos A \), find the value of \(k\).

[BASIC: 430/3/1, 430/3/2, 430/3/3]

35. Prove that \( \sqrt{\frac{\csc A-1}{\csc A+1}}=\sec A-\tan A \).

[BASIC: 430/1/2]

36. Prove that \( (\sin A-\csc A)(\cos A-\sec A) =\frac1{\tan A+\cot A} \).

[BASIC: 430/1/3]

37. Prove that \( \frac{\cos\theta}{1+\sin\theta} +\frac{1+\sin\theta}{\cos\theta} =2\sec\theta \).

[BASIC: 430/2/1]

38. Prove that \( \frac{\cos A-2\cos^3A}{2\sin^3A-\sin A}=\cot A \).

[BASIC: 430/2/3]

39. Prove that \( \frac{1+\csc A}{\csc A} =\frac{\cos^2A}{1-\sin A} \).

[BASIC: 430/3/1]

40. Prove that \( \frac{1+\sec A}{\sec A} =\frac{\sin^2A}{1-\cos A} \).

[BASIC: 430/3/2]

41. Prove that \( \frac{\tan\theta}{1+\cot\theta} +\frac{\cot\theta}{1+\tan\theta} =\tan\theta+\cot\theta-1 \).

[BASIC: 430/3/3]

42. Prove that \( \frac{\sin\theta-\cos\theta+1} {\sin\theta+\cos\theta-1} =\frac1{\sec\theta-\tan\theta} \).

[STD: 30/4/1 Q28]

43. Prove that \( \frac1{\sec x-\tan x}-\frac1{\cos x} =\frac1{\cos x}-\frac1{\sec x+\tan x} \).

[STD: 30/4/2 Q30]

44. Prove that \( \frac{\tan\theta}{1-\cot\theta} +\frac{\cot\theta}{1-\tan\theta} =1+\tan\theta+\cot\theta \).

[STD: 30/5/1 Q28, 30/5/3 Q30]

45. Prove that \( (\sin A+\sec A)^2+(\cos A+\csc A)^2 =(1+\sec A\csc A)^2 \).

[STD: 30/1/3 Q26(b), 30/3/1 Q26(b), 30/3/2 Q28(b)]

46. If \( \sec\theta+\tan\theta=m \), show that \( \frac{m^2-1}{m^2+1}=\sin\theta \).

[STD: 30/4/3 Q31]

47. A car is moving away from the base of a 30 m high tower. The angle of elevation of the top of the tower from the car at an instant, when the car is \(10\sqrt3\) m away from the base, is:

(A) \(30^\circ\)

(B) \(45^\circ\)

(C) \(90^\circ\)

(D) \(60^\circ\)

[STD: 30/1/1 Q11]

48. In the given figure, \(PT\) is a tangent to the circle with centre \(O\) and radius \(r\). If \(\angle POT=45^\circ\), then the length of \(OP\) is:

(A) \(r\sqrt2\)

(B) \(\sqrt{2r}\)

(C) \(2r\)

(D) \(r^2\)

[STD: 30/3/1 Q12]

CHAPTER 8 : INTRODUCTION TO TRIGONOMETRY

1. MULTIPLE CHOICE QUESTIONS (1 MARK)

A. Trigonometric Ratios & Basic Concepts

B. Trigonometric Identities & Simplification

2. ASSERTION–REASON QUESTIONS (1 MARK)

3. VERY SHORT ANSWER QUESTIONS (2 MARKS)

A. Finding Angles

B. Geometrical Proof

C. Evaluation

D. Identity Proofs

4. SHORT ANSWER QUESTIONS (3 MARKS)

A. Prove the Following Identities

5. FIGURE-BASED TRIGONOMETRY