Trigonometry MCQ Questions and Solutions with Explanations

31.
If tan²θ = 1 - e², then the value of secθ + tan³θ.cosecθ is?

A. $${\left( {2 + {e^2}} \right)^2}$$

B. $${\left( {2 - {e^2}} \right)^{\frac{1}{2}}}$$

C. $${\left( {2 + {e^2}} \right)^{\frac{1}{2}}}$$

D. $${\left( {2 - {e^2}} \right)^{\frac{3}{2}}}$$

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32.
If $$\frac{{{\text{cos }}\alpha }}{{{\text{cos }}\beta }} = a$$ and $$\frac{{{\text{sin }}\alpha }}{{{\text{sin }}\beta }} = b{\text{,}}$$ then the value of $${\sin ^2}\beta $$ in terms of a and b is?

A. $$\frac{{{a^2} + 1}}{{{a^2} - {b^2}}}$$

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

C. $$\frac{{{a^2} - 1}}{{{a^2} - {b^2}}}$$

D. $$\frac{{{a^2} - 1}}{{{a^2} + {b^2}}}$$

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33.
sinθ = 0.7 then cosθ, 0 ≤ θ < 90° is?

A. 0.3

B. $$\sqrt {0.49} $$

C. $$\sqrt {0.51} $$

D. $$\sqrt {0.9} $$

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34.
If $${\text{2}}\sin \theta + {\text{cos}}\theta = \frac{7}{3}{\text{,}}$$ then the value of $$\left( {{\text{ta}}{{\text{n}}^2}\theta - {{\sec }^2}\theta } \right)$$ is?

A. 0

B. -1

C. $$\frac{3}{7}$$

D. $$\frac{7}{3}$$

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35.
The value of $${\sec ^2}\theta $$ - $$\frac{{{{\sin }^2}\theta - 2{{\sin }^4}\theta }}{{{\text{2co}}{{\text{s}}^4}\theta - {\text{co}}{{\text{s}}^2}\theta }}$$ is?

A. 1

B. 2

C. -1

D. 0

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36.
$$\sqrt {\frac{{1 + {\text{sin }}\theta }}{{1 - {\text{sin }}\theta }}} $$ + $$\sqrt {\frac{{1 - {\text{sin }}\theta }}{{1 + {\text{sin }}\theta }}} $$ is equal to?

A. 2cosθ

B. 2sinθ

C. 2cotθ

D. 2secθ

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37.
If $${\left( {r\cos \theta - \sqrt 3 } \right)^2}$$ + $${\left( {r\sin \theta - 1} \right)^2}$$ = 0, then the value of $$\frac{{r\tan \theta + \sec \theta }}{{r\sec \theta + \tan \theta }}$$ is equal to?

A. $$\frac{4}{5}$$

B. $$\frac{3}{5}$$

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

D. $$\frac{{\sqrt 5 }}{4}$$

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Answer & Solution

Answer: Option A

Solution:

$$\eqalign{ & {\left( {r\cos \theta - \sqrt 3 } \right)^2} + {\left( {r\sin \theta - 1} \right)^2} = 0 \cr & \Rightarrow {\left( {r\cos \theta - \sqrt 3 } \right)^2} = 0,{\text{ }}{\left( {r\sin \theta - 1} \right)^2} = 0 \cr & \Rightarrow r{\text{ cos}}\theta = \sqrt 3 ........(i) \cr & \Rightarrow r\sin \theta = 1.........(ii) \cr & {\text{Squaring and adding equation (i) and (ii)}} \cr & \Rightarrow {r^2}{\cos ^2}\theta + {r^2}{\sin ^2}\theta = 3 + 1 \cr & \Rightarrow {r^2}\left( {{\text{co}}{{\text{s}}^2}\theta + {{\sin }^2}\theta } \right) = 4 \cr & \Rightarrow {r^2} = 4 \cr & \Rightarrow r = 2 \cr & {\text{tan}}\theta = \frac{{r\sin \theta }}{{r{\text{ cos}}\theta }} = \frac{1}{{\sqrt 3 }}{\text{ and }}r{\text{ cos}}\theta = \sqrt 3 \cr & \cos \theta = \frac{{\sqrt 3 }}{r}, \cr & \sec \theta = \frac{r}{{\sqrt 3 }} \cr & \frac{{r\tan \theta + \sec \theta }}{{r\sec \theta + \tan \theta }} \cr & = \frac{{\frac{r}{{\sqrt 3 }} + \frac{r}{{\sqrt 3 }}}}{{\frac{{{r^2}}}{{\sqrt 3 }} + \frac{1}{{\sqrt 3 }}}} \cr & = \frac{{r\left( {\frac{2}{{\sqrt 3 }}} \right)}}{{\frac{{{r^2} + 1}}{{\sqrt 3 }}}} \cr & = \frac{{2r}}{{{r^2} + 1}} \cr & = \frac{{2 \times 2}}{{{2^2} + 1}} \cr & = \frac{4}{5} \cr & \cr & {\bf{Alternate:}} \cr & r = 2 \cr & {\text{tan}}\theta = \frac{{r\sin \theta }}{{r{\text{ cos}}\theta }} = \frac{1}{{\sqrt 3 }} \cr & \theta = {30^ \circ } \cr & = \frac{{2\tan {{30}^ \circ } + \sec {{30}^ \circ }}}{{2\sec {{30}^ \circ } + \tan {{30}^ \circ }}} \cr & = \frac{{2 \times \frac{1}{{\sqrt 3 }} + \frac{2}{{\sqrt 3 }}}}{{2 \times \frac{2}{{\sqrt 3 }} + \frac{1}{{\sqrt 3 }}}} \cr & = \frac{4}{5} \cr} $$

38.
Let A, B, C, D be the angles of a quadrilateral. If they are concyclic, then the value of cos A + cos B + cos C + cos D is ?

A. 0

B. 1

C. -1

D. 2

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39.
If $$\frac{{{\text{sec}}\theta + {\text{tan}}\theta }}{{{\text{sec}}\theta - {\text{tan}}\theta }} = 2\frac{{51}}{{79}}{\text{,}}$$ then the value of $$\sin \theta $$ is?

A. $$\frac{{91}}{{144}}$$

B. $$\frac{{39}}{{72}}$$

C. $$\frac{{65}}{{144}}$$

D. $$\frac{{35}}{{72}}$$

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40.
If 1 + cos²θ = 3sinθ cosθ, then the integral value of cotθ is $$\left( {0 < \theta < \frac{\pi }{2}} \right) = \,?$$

A. 2

B. 1

C. 3

D. 0

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Trigonometry MCQ Questions and Solutions with Explanations | Aptitude

31. If tan2θ = 1 - e2, then the value of secθ + tan3θ.cosecθ is?

Answer & Solution

32. If $$\frac{{{\text{cos }}\alpha }}{{{\text{cos }}\beta }} = a$$ and $$\frac{{{\text{sin }}\alpha }}{{{\text{sin }}\beta }} = b{\text{,}}$$ then the value of $${\sin ^2}\beta $$ in terms of a and b is?

Answer & Solution

33. sinθ = 0.7 then cosθ, 0 ≤ θ < 90° is?

Answer & Solution

34. If $${\text{2}}\sin \theta + {\text{cos}}\theta = \frac{7}{3}{\text{,}}$$ then the value of $$\left( {{\text{ta}}{{\text{n}}^2}\theta - {{\sec }^2}\theta } \right)$$ is?

Answer & Solution

35. The value of $${\sec ^2}\theta $$ - $$\frac{{{{\sin }^2}\theta - 2{{\sin }^4}\theta }}{{{\text{2co}}{{\text{s}}^4}\theta - {\text{co}}{{\text{s}}^2}\theta }}$$ is?

Answer & Solution

36. $$\sqrt {\frac{{1 + {\text{sin }}\theta }}{{1 - {\text{sin }}\theta }}} $$ + $$\sqrt {\frac{{1 - {\text{sin }}\theta }}{{1 + {\text{sin }}\theta }}} $$ is equal to?

Answer & Solution

37. If $${\left( {r\cos \theta - \sqrt 3 } \right)^2}$$ + $${\left( {r\sin \theta - 1} \right)^2}$$ = 0, then the value of $$\frac{{r\tan \theta + \sec \theta }}{{r\sec \theta + \tan \theta }}$$ is equal to?

Answer & Solution

38. Let A, B, C, D be the angles of a quadrilateral. If they are concyclic, then the value of cos A + cos B + cos C + cos D is ?

Answer & Solution

39. If $$\frac{{{\text{sec}}\theta + {\text{tan}}\theta }}{{{\text{sec}}\theta - {\text{tan}}\theta }} = 2\frac{{51}}{{79}}{\text{,}}$$ then the value of $$\sin \theta $$ is?

Answer & Solution

40. If 1 + cos2θ = 3sinθ cosθ, then the integral value of cotθ is $$\left( {0 < \theta < \frac{\pi }{2}} \right) = \,?$$

Answer & Solution

Read More Section(Trigonometry)

31.
If tan²θ = 1 - e², then the value of secθ + tan³θ.cosecθ is?

32.
If $$\frac{{{\text{cos }}\alpha }}{{{\text{cos }}\beta }} = a$$ and $$\frac{{{\text{sin }}\alpha }}{{{\text{sin }}\beta }} = b{\text{,}}$$ then the value of $${\sin ^2}\beta $$ in terms of a and b is?

33.
sinθ = 0.7 then cosθ, 0 ≤ θ < 90° is?

34.
If $${\text{2}}\sin \theta + {\text{cos}}\theta = \frac{7}{3}{\text{,}}$$ then the value of $$\left( {{\text{ta}}{{\text{n}}^2}\theta - {{\sec }^2}\theta } \right)$$ is?

35.
The value of $${\sec ^2}\theta $$ - $$\frac{{{{\sin }^2}\theta - 2{{\sin }^4}\theta }}{{{\text{2co}}{{\text{s}}^4}\theta - {\text{co}}{{\text{s}}^2}\theta }}$$ is?

36.
$$\sqrt {\frac{{1 + {\text{sin }}\theta }}{{1 - {\text{sin }}\theta }}} $$ + $$\sqrt {\frac{{1 - {\text{sin }}\theta }}{{1 + {\text{sin }}\theta }}} $$ is equal to?

37.
If $${\left( {r\cos \theta - \sqrt 3 } \right)^2}$$ + $${\left( {r\sin \theta - 1} \right)^2}$$ = 0, then the value of $$\frac{{r\tan \theta + \sec \theta }}{{r\sec \theta + \tan \theta }}$$ is equal to?

38.
Let A, B, C, D be the angles of a quadrilateral. If they are concyclic, then the value of cos A + cos B + cos C + cos D is ?

39.
If $$\frac{{{\text{sec}}\theta + {\text{tan}}\theta }}{{{\text{sec}}\theta - {\text{tan}}\theta }} = 2\frac{{51}}{{79}}{\text{,}}$$ then the value of $$\sin \theta $$ is?

40.
If 1 + cos²θ = 3sinθ cosθ, then the integral value of cotθ is $$\left( {0 < \theta < \frac{\pi }{2}} \right) = \,?$$