Class 10 Mathematics — Chapter 8

Introduction to Trigonometry: Question Bank

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A. Multiple Choice Questions (MCQs)

1. In a right-angled triangle, the side opposite the right angle is called the:

A. Adjacent side
B. Opposite side
C. Hypotenuse
D. Base

2. For an acute angle $A$,$$\sin A=$$

A. $\frac{\text{Adjacent}}{\text{Hypotenuse}}$
B. $\frac{\text{Opposite}}{\text{Hypotenuse}}$
C. $\frac{\text{Hypotenuse}}{\text{Opposite}}$
D. $\frac{\text{Opposite}}{\text{Adjacent}}$

3. Which of the following is the reciprocal of $\cos A$?

A. $\cosec A$
B. $\cot A$
C. $\sec A$
D. $\tan A$

4. Which relation is correct?

A. $\tan A=\frac{\cos A}{\sin A}$
B. $\tan A=\sin A\cos A$
C. $\tan A=\frac{\sin A}{\cos A}$
D. $\tan A=\frac1{\sin A}$

5. If $\tan A=\frac43$, then the corresponding sides opposite and adjacent to $A$ can be taken as:

A. $3k,4k$
B. $4k,3k$
C. $4k,5k$
D. $3k,5k$

6. If the opposite and adjacent sides of an acute angle are $5$ and $12$, respectively, its hypotenuse is:

A. 7
B. 13
C. 17
D. 60

7. The value of $\sin45^\circ$ is:

A. $1$
B. $\frac12$
C. $\frac1{\sqrt2}$
D. $\sqrt3$

8. The value of $\tan60^\circ$ is:

A. $\frac1{\sqrt3}$
B. $1$
C. $\frac12$
D. $\sqrt3$

9. Which of the following is not defined?

A. $\sin90^\circ$
B. $\cos0^\circ$
C. $\tan90^\circ$
D. $\cot90^\circ$

10. The value of $\cos60^\circ$ is:

A. $0$
B. $\frac12$
C. $\frac{\sqrt3}{2}$
D. $1$

11. If $\sin A=\frac35$, then $\cos A$, for an acute angle $A$, is:

A. $\frac34$
B. $\frac45$
C. $\frac54$
D. $\frac25$

12. If $\cot A=\frac74$, then $\tan A$ is:

A. $\frac74$
B. $\frac47$
C. $\frac37$
D. $\frac{11}{4}$

13. Which identity is correct?

A. $\sin^2A-\cos^2A=1$
B. $\sin^2A+\cos^2A=1$
C. $\sin A+\cos A=1$
D. $\sin^2A+\tan^2A=1$

14. From $1+\tan^2A=\sec^2A$, we get:

A. $\sec^2A+\tan^2A=1$
B. $\sec^2A-\tan^2A=1$
C. $\tan^2A-\sec^2A=1$
D. $\sec A-\tan A=1$

15. Which identity is valid?

A. $\cosec^2A+\cot^2A=1$
B. $\cosec^2A-\cot^2A=1$
C. $\cot^2A-\cosec^2A=1$
D. $\cosec A-\cot A=1$

16. As an acute angle increases from $0^\circ$ to $90^\circ$:

A. $\sin A$ decreases
B. $\cos A$ increases
C. $\sin A$ increases and $\cos A$ decreases
D. Both remain constant

17. If $\sin A=\sin B$, where $A$ and $B$ are acute angles, then:

A. $A+B=90^\circ$
B. $A=B$
C. $A-B=90^\circ$
D. $A=2B$

18. If $\tan A=1$, then $A$, being acute, is:

A. $30^\circ$
B. $45^\circ$
C. $60^\circ$
D. $90^\circ$

19. If $\sec A=2$, then $\cos A$ is:

A. $2$
B. $\frac12$
C. $\sqrt3$
D. $1$

20. Which ratio is always greater than or equal to $1$ in its defined acute-angle range?

A. $\sin A$
B. $\cos A$
C. $\sec A$
D. $\tan A$

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B. Fill in the Blanks

21. Trigonometry studies the relationship between the ______ and ______ of a triangle.

22. The longest side of a right triangle is called the ______.

23.$$\sin A=\frac{\_\_\_\_\_\_}{\text{hypotenuse}}$$

24.$$\cos A=\frac{\text{adjacent side}}{\_\_\_\_\_\_}$$

25.$$\tan A=\frac{\_\_\_\_\_\_}{\text{adjacent side}}$$

26. $\cosec A$ is the reciprocal of ______.

27. $\sec A$ is the reciprocal of ______.

28. $\cot A$ is the reciprocal of ______.

29.$$\tan A=\frac{\_\_\_\_\_\_}{\cos A}$$

30.$$\cot A=\frac{\cos A}{\_\_\_\_\_\_}$$

31.$$\sin0^\circ=\_\_\_\_\_\_$$

32.$$\cos90^\circ=\_\_\_\_\_\_$$

33.$$\tan45^\circ=\_\_\_\_\_\_$$

34.$$\sin30^\circ=\_\_\_\_\_\_$$

35.$$\cos60^\circ=\_\_\_\_\_\_$$

36.$$\sin^2A+\cos^2A=\_\_\_\_\_\_$$

37.$$1+\tan^2A=\_\_\_\_\_\_$$

38.$$1+\cot^2A=\_\_\_\_\_\_$$

39. $\tan90^\circ$ is ______.

40. $\cosec0^\circ$ is ______.

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C. True or False

41. The value of $\sin A$ can be greater than $1$.

42. The value of $\cos A$ can be greater than $1$ for an acute angle.

43. $\sec A$ is the reciprocal of $\cos A$.

44. $\tan A=\frac{\cos A}{\sin A}$.

45. $\sin45^\circ=\cos45^\circ$.

46. $\tan45^\circ=1$.

47. $\tan90^\circ$ is defined.

48. $\cot0^\circ$ is not defined.

49.$$\sin^2A+\cos^2A=1$$

is a trigonometric identity.

50. $\sin A$ decreases as $A$ increases from $0^\circ$ to $90^\circ$.

51. $\cos A$ decreases from $1$ to $0$ as $A$ increases from $0^\circ$ to $90^\circ$.

52. If two acute angles have equal sine values, the angles are equal.

53. $\cosec A=\sin^{-1}A$.

54. $\tan A=\frac{\sin A}{\cos A}$.

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D. Assertion–Reason Questions

Choose:

A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.

55. Assertion: $\sin A\leq1$ for an acute angle.
Reason: The hypotenuse is the longest side of a right triangle.

56. Assertion: $\tan90^\circ$ is not defined.
Reason: $\tan A=\frac{\sin A}{\cos A}$, and $\cos90^\circ=0$.

57. Assertion: $\sin45^\circ=\cos45^\circ$.
Reason: In a $45^\circ-45^\circ-90^\circ$ triangle, the two perpendicular sides are equal.

58. Assertion: $\sec A\geq1$ for $0^\circ\leq A<90^\circ$.
Reason: $\sec A$ is the reciprocal of $\cos A$, whose value is at most $1$ in this range.

59. Assertion: $\cot0^\circ=0$.
Reason: $\cot A=\frac{\cos A}{\sin A}$.

60. Assertion: The trigonometric ratios of an angle do not change when the size of the similar right triangle changes.
Reason: Corresponding sides of similar triangles are proportional.

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E. Very Short Answer Questions

61. Define trigonometry.

62. What is meant by the hypotenuse of a right triangle?

63. Write the reciprocal of $\tan A$.

64. Write $\tan A$ in terms of $\sin A$ and $\cos A$.

65. Write $\cot A$ in terms of $\sin A$ and $\cos A$.

66. What is the value of $\sin90^\circ$?

67. What is the value of $\cos0^\circ$?

68. What is the value of $\tan45^\circ$?

69. What is the value of $\cot90^\circ$?

70. Name the three fundamental Pythagorean trigonometric identities given in the chapter.

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F. Short Answer / Numerical Questions

71. If $\tan A=\frac34$, find $\sin A$ and $\cos A$.

72. If $\sin A=\frac5{13}$, find $\cos A$ and $\tan A$.

73. Given $15\cot A=8$, find $\sin A$ and $\sec A$.

74. Given $\sec A=\frac{13}{12}$, find all the other trigonometric ratios of $A$.

75. If $\cot A=\frac78$, find $\tan A$.

76. In a right triangle, the sides opposite and adjacent to an acute angle are $8$ cm and $15$ cm. Find all six trigonometric ratios.

77. In a right triangle, hypotenuse $=13$ cm and one perpendicular side $=5$ cm. Find the six ratios corresponding to the angle opposite the $5$ cm side.

78. If $\tan A=1$, verify that:$$2\sin A\cos A=1.$$

79. Evaluate:$$\sin60^\circ\cos30^\circ+ \sin30^\circ\cos60^\circ.$$

80. Evaluate:$$2\tan^245^\circ+\cos^230^\circ-\sin^260^\circ.$$

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G. Identity-Based Questions

81. Prove:$$\sec^2A-\tan^2A=1.$$

82. Prove:$$\cosec^2A-\cot^2A=1.$$

83. If $\tan A=\frac1{\sqrt3}$, find $\sec A,\cos A,\sin A,\cosec A,\cot A$.

84. Express $\cos A,\tan A,\sec A$ in terms of $\sin A$.

85. Prove:$$\sec A(1-\sin A)(\sec A+\tan A)=1.$$

86. Simplify:$$\frac{1+\tan^2A}{\sec^2A}.$$

87. Simplify:$$\frac{\cosec^2A-1}{\cot^2A}.$$

88. If$$\sin A=\frac35,$$

verify that$$\sec^2A-\tan^2A=1.$$

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H. Conceptual & Application-Based Questions

89. Why can the value of $\sin A$ never exceed $1$ in a right triangle?

90. Why are $\cot0^\circ$ and $\cosec0^\circ$ not defined?

91. Why are $\tan90^\circ$ and $\sec90^\circ$ not defined?

92. If two right triangles have the same acute angle but different side lengths, will their sine values be different? Give a reason.

93. If one side and one acute angle of a right triangle are known, can the remaining sides and angles be determined? Explain briefly.

94. A right triangle has one acute angle $30^\circ$ and hypotenuse $10$ cm. Find the side opposite the $30^\circ$ angle.

95. In a right triangle, one acute angle is $60^\circ$ and the side adjacent to it is $5$ cm. Find the hypotenuse.

96. If$$\sin(A-B)=\frac12,\qquad \cos(A+B)=\frac12,$$

with $A>B$ and $0^\circ<A+B\leq90^\circ$, find $A$ and $B$.

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Answer Key

A. MCQs

Q	Answer	Q	Answer
1	C	11	B
2	B	12	B
3	C	13	B
4	C	14	B
5	B	15	B
6	B	16	C
7	C	17	B
8	D	18	B
9	C	19	B
10	B	20	C

B. Fill in the Blanks

21. sides, angles
22. hypotenuse
23. opposite side
24. hypotenuse
25. opposite side
26. $\sin A$
27. $\cos A$
28. $\tan A$
29. $\sin A$
30. $\sin A$
31. $0$
32. $0$
33. $1$
34. $\frac12$
35. $\frac12$
36. $1$
37. $\sec^2A$
38. $\cosec^2A$
39. not defined
40. not defined

C. True/False

41. False
42. False
43. True
44. False
45. True
46. True
47. False
48. True
49. True
50. False
51. True
52. True
53. False
54. True

D. Assertion–Reason

55. A
56. A
57. A
58. A
59. D
60. A

E. Very Short Answers

61. The branch of mathematics dealing with relationships between sides and angles of triangles.
62. The side opposite the $90^\circ$ angle; it is the longest side.

$$\boxed{\cot A}$$

$$\boxed{\tan A=\frac{\sin A}{\cos A}}$$

$$\boxed{\cot A=\frac{\cos A}{\sin A}}$$

66. $1$
67. $1$
68. $1$
69. $0$

$$\boxed{\sin^2A+\cos^2A=1}$$$$\boxed{1+\tan^2A=\sec^2A}$$$$\boxed{1+\cot^2A=\cosec^2A}$$

F. Numerical Answers

$$\boxed{\sin A=\frac35,\quad\cos A=\frac45}$$

$$\boxed{\cos A=\frac{12}{13},\quad\tan A=\frac5{12}}$$

$$\boxed{\sin A=\frac8{17},\quad\sec A=\frac{17}{15}}$$

$$\boxed{\cos A=\frac{12}{13},\ \sin A=\frac5{13},\ \tan A=\frac5{12}}$$$$\boxed{\cosec A=\frac{13}{5},\ \cot A=\frac{12}{5}}$$

$$\boxed{\tan A=\frac87}$$

Hypotenuse:$$\sqrt{8^2+15^2}=17$$

Therefore:$$\boxed{\sin A=\frac8{17},\ \cos A=\frac{15}{17},\ \tan A=\frac8{15}}$$$$\boxed{\cosec A=\frac{17}{8},\ \sec A=\frac{17}{15},\ \cot A=\frac{15}{8}}$$

77. Other side:

$$\sqrt{13^2-5^2}=12$$

For the angle opposite $5$ cm:$$\boxed{\sin A=\frac5{13},\quad \cos A=\frac{12}{13},\quad \tan A=\frac5{12}}$$$$\boxed{\cosec A=\frac{13}{5},\quad \sec A=\frac{13}{12},\quad \cot A=\frac{12}{5}}$$

$$\sin45^\circ=\cos45^\circ=\frac1{\sqrt2}$$

Hence:$$2\sin A\cos A =2\left(\frac1{\sqrt2}\right) \left(\frac1{\sqrt2}\right) =\boxed1$$

$$\frac{\sqrt3}{2}\cdot\frac{\sqrt3}{2} +\frac12\cdot\frac12 =\boxed1$$

$$2(1)^2+\left(\frac{\sqrt3}{2}\right)^2- \left(\frac{\sqrt3}{2}\right)^2 =\boxed2$$

G. Identity Answers

$$\boxed{\sec^2A-\tan^2A=1}$$

$$\boxed{\cosec^2A-\cot^2A=1}$$

$$\boxed{\sec A=\frac2{\sqrt3},\ \cos A=\frac{\sqrt3}{2},\ \sin A=\frac12}$$$$\boxed{\cosec A=2,\quad \cot A=\sqrt3}$$

$$\boxed{\cos A=\sqrt{1-\sin^2A}}$$$$\boxed{\tan A=\frac{\sin A}{\sqrt{1-\sin^2A}}}$$$$\boxed{\sec A=\frac1{\sqrt{1-\sin^2A}}}$$

For the acute-angle context of the chapter, the positive square root is used.

$$\boxed{\text{LHS}=1=\text{RHS}}$$

$$\boxed1$$

Since$$\cosec^2A-1=\cot^2A,$$

the result is:$$\boxed1$$

88. Given:

$$\sin A=\frac35$$

so$$\cos A=\frac45,\qquad \tan A=\frac34,\qquad \sec A=\frac54.$$

Therefore:$$\sec^2A-\tan^2A =\frac{25}{16}-\frac9{16} =\boxed1.$$

H. Application/Conceptual Answers

89. Because the hypotenuse is the longest side, so

$$\frac{\text{opposite}}{\text{hypotenuse}}\leq1.$$

90. Both involve division by $\sin0^\circ=0$.
91. Both involve division by $\cos90^\circ=0$.
92. No. Their sine values remain the same because the triangles are similar and corresponding sides remain proportional.
93. Yes. A known side and an acute angle allow appropriate trigonometric ratios and Pythagoras’ theorem to determine the remaining quantities.

$$\sin30^\circ=\frac{\text{opposite}}{10}$$$$\frac12=\frac{\text{opposite}}{10}$$$$\boxed{\text{opposite}=5\text{ cm}}$$

$$\cos60^\circ=\frac5H$$$$\frac12=\frac5H$$$$\boxed{H=10\text{ cm}}$$

$$\sin(A-B)=\frac12 \Rightarrow A-B=30^\circ$$

and$$\cos(A+B)=\frac12 \Rightarrow A+B=60^\circ.$$

Adding:$$2A=90^\circ\Rightarrow\boxed{A=45^\circ}$$

Thus: B=15∘​