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TRIGONOMETRIC FUNCTIONS-Objective Questions for Entrance Exams

The following set of 50 objective-type questions on Trigonometric Functions has been meticulously curated to reflect the conceptual depth, algebraic rigor, and analytical patterns repeatedly observed in major competitive entrance examinations such as IIT-JEE (old), JEE Main, JEE Advanced, NEET (AIPMT), AIIMS, BITSAT, KVPY, various Olympiads, and State Engineering Entrance Exams. Each question is framed to test not only formula recall but also conceptual clarity, domain–range awareness, inverse trigonometric reasoning, transformation techniques, and maximum–minimum analysis—skills that are indispensable for high-stakes examinations. The progression of questions moves from standard identity-based problems to multi-solution equations and Olympiad-level reasoning, mirroring real exam difficulty gradients. Detailed explanations accompany every question to reinforce correct reasoning, eliminate common misconceptions, and strengthen problem-solving strategies. This collection serves equally well as a practice set, revision module, and conceptual diagnostic tool for serious aspirants aiming for excellence in competitive mathematics.

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Exercise

TRIGONOMETRIC FUNCTIONS

by Academia Aeternum

1. If \(\sin \theta + \cos \theta = \sqrt{2}\), then \(\theta\) equals
(Exam: IIT-JEE Year: 1998)
2. The value of \(\tan 75^\circ\) is
(Exam: JEE Main Year: 2015)
3. If \(\sin \theta = \frac{3}{5}\) and \(\theta\) lies in the first quadrant, then \(\cos \theta\) equals
(Exam: NEET Year: 2019)
4. The principal value of \(\sin^{-1}(-\frac{1}{2})\) is
(Exam: JEE Advanced Year: 2014)
5. The value of \(\sin 18^\circ\) is
(Exam: Olympiad Year: 2002)
6. If \(\tan \theta = -1\) and \(\theta \in (0, 2\pi)\), then \(\theta\) equals
(Exam: BITSAT Year: 2017)
7. The maximum value of \(\sin \theta \cos \theta\) is
(Exam: JEE Main Year: 2016)
8. If \(\sin \theta = \cos \theta\), then \(\theta\) equals
(Exam: State Engg. Exam Year: 2013)
9. The value of \(\cos 2\theta\) when \(\sin \theta = \frac{1}{2}\) is
(Exam: NEET Year: 2020)
10. The number of solutions of \(\sin x = \frac{1}{2}\) in \([0, 2\pi]\) is
(Exam: JEE Advanced Year: 2011)
11. The value of \(\sin 20^\circ \cos 40^\circ + \cos 20^\circ \sin 40^\circ\) is
(Exam: IIT-JEE Year: 2001)
12. The principal value of \(\tan^{-1}(\sqrt{3})\) is
(Exam: JEE Main Year: 2014)
13. If \(\cos(\theta-\frac{\pi}{6})=\frac{\sqrt{3}}{2}\), then \(\theta\) equals
(Exam: NEET Year: 2018)
14. The maximum value of \(3\sin x + 4\cos x\) is
(Exam: BITSAT Year: 2016)
15. The domain of \(\sin^{-1}(2x-1)\) is
(Exam: JEE Main Year: 2017)
16. The number of solutions of \(\cos x = \sin x\) in \([0,2\pi]\) is
(Exam: IIT-JEE Year: 1996)
17. If \(\sin \theta + \cos \theta = 1\), then \(\sin \theta \cos \theta\) equals
(Exam: Olympiad Year: 2005)
18. The value of \(\tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3\) is
(Exam: JEE Advanced Year: 2013)
19. If \(\sin x = \sin 3x\) for \(x\in(0,\pi)\), then \(x\) equals
(Exam: NEET Year: 2021)
20. The range of \(y=\cos^{-1}(\sin x)\) is
(Exam: JEE Main Year: 2019)
21. The value of \(\sin 2\theta\) when \(\tan \theta=1\) is
(Exam: State Engg. Exam Year: 2012)
22. The number of solutions of \(\tan x = \sqrt{3}\) in \([0,2\pi]\) is
(Exam: JEE Main Year: 2018)
23. If \(\cos x = -\frac{1}{2}\) and \(x\in(0,2\pi)\), then \(x\) equals
(Exam: NEET Year: 2016)
24. The minimum value of \(\sin x + \cos x\) is
(Exam: IIT-JEE Year: 1999)
25. The value of \(\cos(\sin^{-1}\frac{3}{5})\) is
(Exam: BITSAT Year: 2015)
26. The equation \(\sin x + \cos x = 0\) has solution
(Exam: JEE Advanced Year: 2010)
27. If \(\sin \theta=\cos \phi\), then \(\theta+\phi\) equals
(Exam: Olympiad Year: 2008)
28. The value of \(\sin(\cos^{-1}x)\) is
(Exam: JEE Main Year: 2020)
29. The general solution of \(\sin x=0\) is
(Exam: State Engg. Exam Year: 2011)
30. The value of \(\cos 36^\circ\) is
(Exam: Olympiad Year: 2004)
31. If \(\tan x + \sec x = 2\), then \(x\) equals
(Exam: JEE Advanced Year: 2012)
32. The number of solutions of \(\sin 2x = 0\) in \([0,2\pi]\) is
(Exam: NEET Year: 2017)
33. If \(\cos x = \sin 2x\), then one solution is
(Exam: JEE Main Year: 2016)
34. The range of \(y=\sin^{-1}(\cos x)\) is
(Exam: IIT-JEE Year: 2003)
35. The value of \(\tan(\sin^{-1}\frac{1}{2})\) is
(Exam: BITSAT Year: 2018)
36. If \(\sin x + \sin 3x = 0\), then \(x\) equals
(Exam: JEE Advanced Year: 2009)
37. The minimum value of \(\sec x + \csc x\) is
(Exam: Olympiad Year: 2007)
38. If \(\sin \theta = -\frac{1}{2}\), then \(\theta\) lies in
(Exam: NEET Year: 2015)
39. The value of \(\cos(\tan^{-1}1)\) is
(Exam: JEE Main Year: 2021)
40. The number of solutions of \(\cos 2x = 1\) in \([0,2\pi]\) is
(Exam: State Engg. Exam Year: 2014)
41. If \(\tan \theta = \frac{5}{12}\), then \(\sin \theta\) equals
(Exam: NEET Year: 2022)
42. The value of \(\sin(\cos^{-1}\frac{1}{2})\) is
(Exam: JEE Main Year: 2018)
43. The equation \(\tan x = \tan 2x\) has solution
(Exam: IIT-JEE Year: 2000)
44. The maximum value of \(\sin^2 x + \cos x\) is
(Exam: Olympiad Year: 2006)
45. The range of \(y=\tan^{-1}(\sin x)\) is
(Exam: JEE Main Year: 2020)
46. The value of \(\cos(\sin^{-1}x)+\sin(\cos^{-1}x)\) is
(Exam: JEE Advanced Year: 2015)
47. If \(\sin x = \cos 3x\), then \(x\) equals
(Exam: NEET Year: 2019)
48. The minimum value of \(\sin x - \cos x\) is
(Exam: IIT-JEE Year: 1997)
49. The number of solutions of \(\sin x = \cos x\) in \((0,\pi)\) is
(Exam: JEE Main Year: 2022)
50. The value of \(\tan(\cos^{-1}\frac{3}{5})\) is
(Exam: Olympiad Year: 2009)

Frequently Asked Questions

Trigonometrical functions are functions that relate an angle to ratios of sides of a right-angled triangle or to coordinates on the unit circle

Because each angle corresponds to a unique real value of sine, cosine, tangent, etc

An angle measured in radians can take any real value, positive or negative

Radian is the angle subtended at the center of a circle by an arc equal in length to the radius

There are p radians in 180 degrees

The domain of sin x and cos x is all real numbers

The range is from -1 to 1 inclusive

All real numbers except odd multiples of \(\pi/2\)

Periodicity means the function repeats its values after a fixed interval

The period is \(2\pi\)

The period is \(\pi)

Identities that hold true for all permissible values of \(x\), such as \(\sin^2 x + \cos^2 x = 1\)

Identities that relate trigonometric functions as reciprocals of each other

Identities expressing tan x and cot x as ratios of sine and cosine

Identities derived from the Pythagorean theorem involving sin, cos, and tan

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