Trigonometry MCQ Questions and Solutions with Explanations

91.
If xtan60° + cos45° = sec45°, then the value of x² + 1 is?

A. $$\frac{6}{7}$$

B. $$\frac{7}{6}$$

C. $$\frac{5}{6}$$

D. $$\frac{6}{5}$$

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92.
x, y be two acute angles, x + y < 90° and sin(2x - 20°) = cos(2y + 20°), the value of tan(x + y) is?

A. $$\sqrt 3 $$

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

C. 1

D. $${\text{2}} + \sqrt 2 $$

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93.
If a² sec²x - b² tan²x = c², then the value of sec²x + tan²x is equal to (assume b² ≠ a²)

A. $$\frac{{{b^2} - {a^2} + 2{c^2}}}{{{b^2} + {a^2}}}$$

B. $$\frac{{{b^2} + {a^2} - 2{c^2}}}{{{b^2} - {a^2}}}$$

C. $$\frac{{{b^2} - {a^2} - 2{c^2}}}{{{b^2} + {a^2}}}$$

D. $$\frac{{{b^2} - {a^2}}}{{{b^2} + {a^2} + 2{c^2}}}$$

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94.
If tan⁴θ + tan²θ = 1, then the value of cos⁴θ + cos²θ is?

A. 2

B. 0

C. 1

D. -1

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95.
The value of 8(sin⁶θ + cos⁶θ) - 12(sin⁴θ + cos⁴θ) is equal to?

A. 20

B. -20

C. -4

D. 4

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96.
If secA = $$\frac{{17}}{8},$$ given that A < 90°, what is the value of the following?
$$\frac{{34\sin A + 15\cot A}}{{68\cos A - 16\tan A}}$$

A. 23

B. 19

C. 30

D. 38

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97.
If A is an acute angle, the simplified form of $$\frac{{\cos \left( {\pi - A} \right).\cot \left( {\frac{\pi }{2} + A} \right)\cos \left( { - A} \right)}}{{\tan \left( {\pi + A} \right)\tan \left( {\frac{{3\pi }}{2} + A} \right)\sin \left( {2\pi - A} \right)}}\,{\text{is:}}$$

A. cos²A

B. sinA

C. sin²A

D. cosA

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98.
If tan²A + 2tanA - 63 = 0, given that 0 < A < $$\frac{\pi }{2}$$ what is the value of (2sinA + 5cosA)?

A. $$19\sqrt {50} $$

B. $$15\sqrt {50} $$

C. $$\frac{{19}}{{\sqrt {50} }}$$

D. $$\frac{{15}}{{\sqrt {50} }}$$

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99.
If $$\frac{{{{\sin }^2}\phi - 3\sin \phi + 2}}{{{{\cos }^2}\phi }} = 1,$$ where 0° < $$\phi $$ < 90° then what is the value of (cos2$$\phi $$ - sin3$$\phi $$ + cosec2$$\phi $$)?

A. $$\frac{{ - 3 - 4\sqrt 3 }}{6}$$

B. $$\frac{{ - 3 + 4\sqrt 3 }}{6}$$

C. $$\frac{{3 + 4\sqrt 3 }}{6}$$

D. $$\frac{{3 - 4\sqrt 3 }}{6}$$

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Answer & Solution

Answer: Option B

Solution:

$$\eqalign{ & \frac{{{{\sin }^2}\phi - 3\sin \phi + 2}}{{{{\cos }^2}\phi }} = 1 \cr & \Rightarrow {\sin ^2}\phi - 3\sin \phi + 2 = {\cos ^2}\phi \cr & \Rightarrow {\sin ^2}\phi - 3\sin \phi + 2 = 1 - {\sin ^2}\phi \cr & \Rightarrow 2{\sin ^2}\phi - 3\sin \phi + 1 = 0 \cr & \Rightarrow 2{\sin ^2}\phi - 2\sin \phi - \sin \phi + 1 = 0 \cr & \Rightarrow 2\sin \phi \left( {\sin \phi - 1} \right) - 1\left( {\sin \phi - 1} \right) = 0 \cr & \Rightarrow \left( {\sin \phi - 1} \right)\left( {2\sin \phi - 1} \right) = 0 \cr & \therefore \sin \phi = 1\,\,\left( {{\text{not valid}}} \right) \cr & {\text{and }}\sin \phi = \frac{1}{2} \cr & \Rightarrow \sin \phi = \sin {30^ \circ } \cr & \therefore \phi = {30^ \circ } \cr & \therefore \cos 2\phi - \sin 3\phi + {\text{cosec }}2\phi \cr & = \cos {60^ \circ } - \sin {90^ \circ } + {\text{cosec }}{60^ \circ } \cr & = \frac{1}{2} - 1 + \frac{2}{{\sqrt 3 }} \cr & = \frac{2}{{\sqrt 3 }} - \frac{1}{2} \cr & = \frac{{\left( {4 - \sqrt 3 } \right)}}{{2\sqrt 3 }} \times \frac{{\sqrt 3 }}{{\sqrt 3 }} \cr & = \frac{{4\sqrt 3 - 3}}{6} \cr & \cr & {\bf{Alternative:}} \cr & {\text{Put }}\phi = {30^ \circ } \cr & {\text{L}}{\text{.H}}{\text{.S}}{\text{.}} = \frac{{{{\sin }^2}{{30}^ \circ } - 3\sin {{30}^ \circ } + 2}}{{{{\cos }^2}{{30}^ \circ }}} \cr & = \frac{{\frac{1}{4} - \frac{3}{2} + 2}}{{\frac{3}{4}}} \cr & = \frac{{1 - 6 + 8}}{3} \cr & = \frac{3}{3} \cr & = 1 = {\text{R}}{\text{.H}}{\text{.S}}{\text{.}} \cr & \therefore \cos 2\phi - \sin 3\phi + {\text{cosec }}2\phi \cr & = \cos {60^ \circ } - \sin {90^ \circ } + {\text{cosec }}{60^ \circ } \cr & = \frac{1}{2} - 1 + \frac{2}{{\sqrt 3 }} \cr & = - \frac{1}{2} + \frac{2}{{\sqrt 3 }} \cr & = \frac{{\left( { - \sqrt 3 + 4} \right)}}{{2\sqrt 3 }} \times \frac{{\sqrt 3 }}{{\sqrt 3 }} \cr & = \frac{{ - 3 + 4\sqrt 3 }}{6} \cr} $$

100.
If sinθ = √3cosθ, 0°< θ < 90°, then the value of 2sin²θ + 6sec²θ + sinθ secθ + cosecθ is

A. $$\frac{{33 + 10\sqrt 3 }}{6}$$

B. $$\frac{{19 + 10\sqrt 3 }}{3}$$

C. $$\frac{{19 + 10\sqrt 3 }}{6}$$

D. $$\frac{{33 + 10\sqrt 3 }}{3}$$

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Trigonometry MCQ Questions and Solutions with Explanations | Aptitude

91. If xtan60° + cos45° = sec45°, then the value of x2 + 1 is?

Answer & Solution

92. x, y be two acute angles, x + y < 90° and sin(2x - 20°) = cos(2y + 20°), the value of tan(x + y) is?

Answer & Solution

93. If a2 sec2x - b2 tan2x = c2, then the value of sec2x + tan2x is equal to (assume b2 ≠ a2)

Answer & Solution

94. If tan4θ + tan2θ = 1, then the value of cos4θ + cos2θ is?

Answer & Solution

95. The value of 8(sin6θ + cos6θ) - 12(sin4θ + cos4θ) is equal to?

Answer & Solution

96. If secA = $$\frac{{17}}{8},$$ given that A < 90°, what is the value of the following? $$\frac{{34\sin A + 15\cot A}}{{68\cos A - 16\tan A}}$$

Answer & Solution

97. If A is an acute angle, the simplified form of $$\frac{{\cos \left( {\pi - A} \right).\cot \left( {\frac{\pi }{2} + A} \right)\cos \left( { - A} \right)}}{{\tan \left( {\pi + A} \right)\tan \left( {\frac{{3\pi }}{2} + A} \right)\sin \left( {2\pi - A} \right)}}\,{\text{is:}}$$

Answer & Solution

98. If tan2A + 2tanA - 63 = 0, given that 0 < A < $$\frac{\pi }{2}$$ what is the value of (2sinA + 5cosA)?

Answer & Solution

99. If $$\frac{{{{\sin }^2}\phi - 3\sin \phi + 2}}{{{{\cos }^2}\phi }} = 1,$$ where 0° < $$\phi $$ < 90° then what is the value of (cos2$$\phi $$ - sin3$$\phi $$ + cosec2$$\phi $$)?

Answer & Solution

100. If sinθ = √3cosθ, 0°< θ < 90°, then the value of 2sin2θ + 6sec2θ + sinθ secθ + cosecθ is

Answer & Solution

Read More Section(Trigonometry)

91.
If xtan60° + cos45° = sec45°, then the value of x² + 1 is?

92.
x, y be two acute angles, x + y < 90° and sin(2x - 20°) = cos(2y + 20°), the value of tan(x + y) is?

93.
If a² sec²x - b² tan²x = c², then the value of sec²x + tan²x is equal to (assume b² ≠ a²)

94.
If tan⁴θ + tan²θ = 1, then the value of cos⁴θ + cos²θ is?

95.
The value of 8(sin⁶θ + cos⁶θ) - 12(sin⁴θ + cos⁴θ) is equal to?

96.
If secA = $$\frac{{17}}{8},$$ given that A < 90°, what is the value of the following?
$$\frac{{34\sin A + 15\cot A}}{{68\cos A - 16\tan A}}$$

97.
If A is an acute angle, the simplified form of $$\frac{{\cos \left( {\pi - A} \right).\cot \left( {\frac{\pi }{2} + A} \right)\cos \left( { - A} \right)}}{{\tan \left( {\pi + A} \right)\tan \left( {\frac{{3\pi }}{2} + A} \right)\sin \left( {2\pi - A} \right)}}\,{\text{is:}}$$

98.
If tan²A + 2tanA - 63 = 0, given that 0 < A < $$\frac{\pi }{2}$$ what is the value of (2sinA + 5cosA)?

99.
If $$\frac{{{{\sin }^2}\phi - 3\sin \phi + 2}}{{{{\cos }^2}\phi }} = 1,$$ where 0° < $$\phi $$ < 90° then what is the value of (cos2$$\phi $$ - sin3$$\phi $$ + cosec2$$\phi $$)?

100.
If sinθ = √3cosθ, 0°< θ < 90°, then the value of 2sin²θ + 6sec²θ + sinθ secθ + cosecθ is