Trigonometry MCQ Questions and Solutions with Explanations

61.
If $$x = \frac{{2\sin \theta }}{{1 + \cos \theta + \sin \theta }},$$ then the value of $$\frac{{1 - \cos \theta + \sin \theta }}{{1 + \sin \theta }}$$ is

A. $$\frac{x}{{1 + x}}$$

B. $$x$$

C. $$\frac{1}{x}$$

D. $$\frac{{1 + x}}{x}$$

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62.
If tan4θ = cot(40° - 2θ), then θ is equal to:

A. 20°

B. 25°

C. 35°

D. 30°

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63.
What is the value of $$\frac{{32{{\cos }^6}x - 48{{\cos }^4}x + 18{{\cos }^2}x - 1}}{{4\sin x\,\cos x\,\sin \left( {60 - x} \right)\cos \left( {60 - x} \right)\sin \left( {60 + x} \right)\cos \left( {60 + x} \right)}}?$$

A. 4tan6x

B. 4cot6x

C. 8cot6x

D. 8tan6x

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64.
$$\left( {\frac{{{{\tan }^3}\theta }}{{{{\sec }^2}\theta }} + \frac{{{{\cot }^3}\theta }}{{{\text{cose}}{{\text{c}}^2}\theta }} + 2\sin \theta \cos \theta } \right)$$ ÷ (1 + cosec²θ + tan²θ), 0° < θ < 90°, is equal to:

A. cosecθ

B. cosecθsecθ

C. sinθcosθ

D. secθ

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65.
The expression $$\frac{{{{\left( {1 - \sin \theta + \cos \theta } \right)}^2}\left( {1 - \cos \theta } \right){{\sec }^3}\theta \,{\text{cose}}{{\text{c}}^2}\theta }}{{\left( {\sec \theta - \tan \theta } \right)\left( {\tan \theta + \cot \theta } \right)}},$$ 0° < θ < 90°, is equal to:

A. 2tanθ

B. cotθ

C. sinθ

D. 2cosθ

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Answer & Solution

Answer: Option A

Solution:

$$\eqalign{ & \frac{{{{\left( {1 - \sin \theta + \cos \theta } \right)}^2}\left( {1 - \cos \theta } \right){{\sec }^3}\theta {\text{cose}}{{\text{c}}^2}\theta }}{{\left( {\sec \theta - \tan \theta } \right)\left( {\tan \theta + \cot \theta } \right)}} \cr & {\text{put }}\theta = {30^ \circ } \cr & = \frac{{{{\left( {1 - \sin {{30}^ \circ } + \cos {{30}^ \circ }} \right)}^2}\left( {1 - \cos {{30}^ \circ }} \right){{\sec }^3}{{30}^ \circ }{\text{cose}}{{\text{c}}^2}{{30}^ \circ }}}{{\left( {\sec {{30}^ \circ } - \tan {{30}^ \circ }} \right)\left( {\tan {{30}^ \circ } + \cot {{30}^ \circ }} \right)}} \cr & = \frac{{{{\left( {1 - \frac{1}{2} + \frac{{\sqrt 3 }}{2}} \right)}^2}\left( {1 - \frac{{\sqrt 3 }}{2}} \right){{\left( {\frac{2}{{\sqrt 3 }}} \right)}^3}{{\left( 2 \right)}^2}}}{{\left( {\frac{2}{{\sqrt 3 }} - \frac{1}{{\sqrt 3 }}} \right)\left( {\frac{1}{{\sqrt 3 }} + \sqrt 3 } \right)}} \cr & = \frac{{{{\left( {\frac{{1 + \sqrt 3 }}{2}} \right)}^2}\left( {\frac{{2 - \sqrt 3 }}{2}} \right){{\left( {\frac{2}{{\sqrt 3 }}} \right)}^3}{{\left( 2 \right)}^2}}}{{\left( {\frac{1}{{\sqrt 3 }}} \right)\left( {\frac{{1 + 3}}{{\sqrt 3 }}} \right)}} \cr & = \frac{{\frac{{4 + 2\sqrt 3 }}{4} \times \frac{{2 - \sqrt 3 }}{2} \times \frac{8}{{3\sqrt 3 }} \times 4}}{{\left( {\frac{1}{{\sqrt 3 }}} \right)\left( {\frac{4}{{\sqrt 3 }}} \right)}} \cr & = \frac{{\left[ {{2^2} - {{\left( {\sqrt 3 } \right)}^2}} \right] \times \frac{8}{{3\sqrt 3 }}}}{{\frac{4}{3}}} \cr & = \frac{{\frac{8}{{3\sqrt 3 }}}}{{\frac{4}{3}}} \cr & = \frac{2}{{\sqrt 3 }} \cr & {\text{Option A: }}2\tan {30^ \circ } = \frac{2}{{\sqrt 3 }} \cr & {\text{Option B: }}\cot {30^ \circ } = \sqrt 3 \cr & {\text{Option C: }}\sin {30^ \circ } = \frac{1}{2} \cr & {\text{Option D: }}2\cos {30^ \circ } = 2 \times \frac{{\sqrt 3 }}{2} = \sqrt 3 \cr & {\text{Only option A is answer}}{\text{.}} \cr} $$

66.
If $$\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{{\text{k}} + 1}}{{{\text{k}} - 1}},$$ then k = ?

A. cosecθ

B. secθ

C. cosθ

D. sinθ

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67.
If $$\sin \theta = \frac{{{p^2} - 1}}{{{p^2} + 1}},$$ then cosθ is equal to:

A. $$\frac{{2p}}{{1 + {p^2}}}$$

B. $$\frac{p}{{{p^2} - 1}}$$

C. $$\frac{p}{{1 + {p^2}}}$$

D. $$\frac{{2p}}{{{p^2} - 1}}$$

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68.
If $$\left( {\frac{{\tan \theta - \sec \theta + 1}}{{\tan \theta + \sec \theta - 1}}} \right)\sec \theta = \frac{1}{{\text{k}}},$$ then k = . . . . . . . .

A. 1 + sinθ

B. 1 - cosθ

C. 1 + cosθ

D. 1 - sinθ

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69.
The value of m[sinθ + 2cos²∅ + 3sinθ + 4cos²∅ + . . . . . . . . + 18cos²∅] is a perfect square of an integer, θ = 30°, ∅ = 45° and 150 ≤ m ≤ 180. Find the value of m.

A. 161

B. 176

C. 152

D. 168

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70.
If 1 + sin²θ - 3sinθcosθ = 0, then the value of cotθ is:

A. 2

B. $$\frac{1}{3}$$

C. $$\frac{1}{2}$$

D. 0

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Trigonometry MCQ Questions and Solutions with Explanations | Aptitude

61. If $$x = \frac{{2\sin \theta }}{{1 + \cos \theta + \sin \theta }},$$ then the value of $$\frac{{1 - \cos \theta + \sin \theta }}{{1 + \sin \theta }}$$ is

Answer & Solution

62. If tan4θ = cot(40° - 2θ), then θ is equal to:

Answer & Solution

63. What is the value of $$\frac{{32{{\cos }^6}x - 48{{\cos }^4}x + 18{{\cos }^2}x - 1}}{{4\sin x\,\cos x\,\sin \left( {60 - x} \right)\cos \left( {60 - x} \right)\sin \left( {60 + x} \right)\cos \left( {60 + x} \right)}}?$$

Answer & Solution

64. $$\left( {\frac{{{{\tan }^3}\theta }}{{{{\sec }^2}\theta }} + \frac{{{{\cot }^3}\theta }}{{{\text{cose}}{{\text{c}}^2}\theta }} + 2\sin \theta \cos \theta } \right)$$ ÷ (1 + cosec2θ + tan2θ), 0° < θ < 90°, is equal to:

Answer & Solution

65. The expression $$\frac{{{{\left( {1 - \sin \theta + \cos \theta } \right)}^2}\left( {1 - \cos \theta } \right){{\sec }^3}\theta \,{\text{cose}}{{\text{c}}^2}\theta }}{{\left( {\sec \theta - \tan \theta } \right)\left( {\tan \theta + \cot \theta } \right)}},$$ 0° < θ < 90°, is equal to:

Answer & Solution

66. If $$\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{{\text{k}} + 1}}{{{\text{k}} - 1}},$$ then k = ?

Answer & Solution

67. If $$\sin \theta = \frac{{{p^2} - 1}}{{{p^2} + 1}},$$ then cosθ is equal to:

Answer & Solution

68. If $$\left( {\frac{{\tan \theta - \sec \theta + 1}}{{\tan \theta + \sec \theta - 1}}} \right)\sec \theta = \frac{1}{{\text{k}}},$$ then k = . . . . . . . .

Answer & Solution

69. The value of m[sinθ + 2cos2∅ + 3sinθ + 4cos2∅ + . . . . . . . . + 18cos2∅] is a perfect square of an integer, θ = 30°, ∅ = 45° and 150 ≤ m ≤ 180. Find the value of m.

Answer & Solution

70. If 1 + sin2θ - 3sinθcosθ = 0, then the value of cotθ is:

Answer & Solution

Read More Section(Trigonometry)

61.
If $$x = \frac{{2\sin \theta }}{{1 + \cos \theta + \sin \theta }},$$ then the value of $$\frac{{1 - \cos \theta + \sin \theta }}{{1 + \sin \theta }}$$ is

62.
If tan4θ = cot(40° - 2θ), then θ is equal to:

63.
What is the value of $$\frac{{32{{\cos }^6}x - 48{{\cos }^4}x + 18{{\cos }^2}x - 1}}{{4\sin x\,\cos x\,\sin \left( {60 - x} \right)\cos \left( {60 - x} \right)\sin \left( {60 + x} \right)\cos \left( {60 + x} \right)}}?$$

64.
$$\left( {\frac{{{{\tan }^3}\theta }}{{{{\sec }^2}\theta }} + \frac{{{{\cot }^3}\theta }}{{{\text{cose}}{{\text{c}}^2}\theta }} + 2\sin \theta \cos \theta } \right)$$ ÷ (1 + cosec²θ + tan²θ), 0° < θ < 90°, is equal to:

65.
The expression $$\frac{{{{\left( {1 - \sin \theta + \cos \theta } \right)}^2}\left( {1 - \cos \theta } \right){{\sec }^3}\theta \,{\text{cose}}{{\text{c}}^2}\theta }}{{\left( {\sec \theta - \tan \theta } \right)\left( {\tan \theta + \cot \theta } \right)}},$$ 0° < θ < 90°, is equal to:

66.
If $$\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{{\text{k}} + 1}}{{{\text{k}} - 1}},$$ then k = ?

67.
If $$\sin \theta = \frac{{{p^2} - 1}}{{{p^2} + 1}},$$ then cosθ is equal to:

68.
If $$\left( {\frac{{\tan \theta - \sec \theta + 1}}{{\tan \theta + \sec \theta - 1}}} \right)\sec \theta = \frac{1}{{\text{k}}},$$ then k = . . . . . . . .

69.
The value of m[sinθ + 2cos²∅ + 3sinθ + 4cos²∅ + . . . . . . . . + 18cos²∅] is a perfect square of an integer, θ = 30°, ∅ = 45° and 150 ≤ m ≤ 180. Find the value of m.

70.
If 1 + sin²θ - 3sinθcosθ = 0, then the value of cotθ is: