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Trigonometry

Trigonometric Ratio & Identities: JEE Main & Advanced PYQs

381 previous year questions on Trigonometric Ratio & Identities from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 1-50 of 381

  1. All possible values of $\theta\in[0,2\pi]$ for which $\sin2\theta+\tan2\theta>0$ lie in :
  2. If $e^{\left(\cos^{2}x+\cos^{4}x+\cos^{6}x+\ldots\infty\right)\log_e2}$ satisfies the equation $t^{2}-9t+8=0$, then the value of …
  3. A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point $A$, with uniform speed. At that point, angle of depression of the …
  4. The number of solutions of $\sin^{7}x+\cos^{7}x=1$, $x\in[0,4\pi]$ is equal to :
  5. If $0<x,y<\pi$ and $\cos x+\cos y-\cos(x+y)=\frac{3}{2}$, then $\sin x+\cos y$ is equal to :
  6. The number of solutions of the equation $x+2\tan x=\frac{\pi}{2}$ in the interval $[0,2\pi]$ is :
  7. The angle of elevation of a jet plane from a point $A$ on the ground is $60^{\circ}$. After a flight of $20$ seconds at the speed of $432$ km/hour, the angle of …
  8. Two vertical poles are $150$ m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds …
  9. Let $P=\{\theta\in[0,4\pi]:\tan^{2}\theta\neq 1\}$ and $S=\{a\in\mathbf{Z}:2(\cos^{8}\theta-\sin^{8}\theta)\sec 2\theta=a^{2},\theta\in P\}$. Then $n(S)$ is :
  10. A pole stands vertically inside a triangular park $ABC$. Let the angle of elevation of the top of the pole from each corner of the park be $\frac{\pi}{3}$. If the radius …
  11. If $15\sin^{4}\alpha+10\cos^{4}\alpha=6$, for some $\alpha\in\mathbf{R}$, then the value of $27\sec^{6}\alpha+8\operatorname{cosec}^{6}\alpha$ is equal to :
  12. The maximum value of $(\cos\theta_1)\cdot(\cos\theta_2)\cdot(\cos\theta_3)\cdot\ldots\cdot(\cos\theta_{10})$ under the restrictions …
  13. The number of solutions of the equation $\cos2\theta\cos\frac{\theta}{2}+\cos\frac{5\theta}{2}=2\cos^{3}\frac{5\theta}{2}$ in $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ …
  14. If $2\sin^{3}x+\sin2x\cos x+4\sin x-4=0$ has exactly $3$ solutions in the interval $\left[0,\frac{n\pi}{2}\right]$, $n\in\mathbf{N}$, then the roots of the equation …
  15. If $\theta\in[-2\pi,2\pi]$, then the number of solutions of $2\sqrt{2}\cos^{2}\theta+\left(2-\sqrt{6}\right)\cos\theta-\sqrt{3}=0$, is equal to :
  16. Let $AB$ and $PQ$ be two vertical poles, $160$ m apart from each other. Let $C$ be the middle point of $B$ and $Q$, which are feet of these two poles. Let …
  17. If $\theta\in\left[-\frac{7\pi}{6},\frac{4\pi}{3}\right]$, then the number of solutions of …
  18. The number of elements in the set $S=\left\{\theta\in[0,2\pi]:3\cos^{4}\theta-5\cos^{2}\theta-2\sin^{6}\theta+2=0\right\}$ is
  19. The number of solutions of the equation $\cos\left(x+\frac{\pi}{3}\right)\cos\left(\frac{\pi}{3}-x\right)=\frac{1}{4}\cos^{2}2x$, $x\in[-3\pi,3\pi]$ is:
  20. A vertical pole fixed to the horizontal ground is divided in the ratio $3:7$ by a mark on it with lower part shorter than the upper part. If the two parts subtend equal …
  21. Let $S=\left\{x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right):9^{1-\tan^{2}x}+9^{\tan^{2}x}=10\right\}$ and $\beta=\sum_{x\in S}\tan^{2}\left(\frac{x}{3}\right)$, then …
  22. The number of solutions of the equation $2x+3\tan x=\pi$, $x\in[-2\pi,2\pi]-\left\{\pm\frac{\pi}{2},\pm\frac{3\pi}{2}\right\}$ is:
  23. The value of $\cos\left(\frac{2\pi}{7}\right)+\cos\left(\frac{4\pi}{7}\right)+\cos\left(\frac{6\pi}{7}\right)$ is equal to :
  24. Let $\frac{\sin A}{\sin B}=\frac{\sin(A-C)}{\sin(C-B)}$, where $A,B,C$ are angles of a triangle $ABC$. If the lengths of the sides opposite these angles are $a,b,c$ …
  25. $96\cos\frac{\pi}{33}\cos\frac{2\pi}{33}\cos\frac{4\pi}{33}\cos\frac{8\pi}{33}\cos\frac{16\pi}{33}$ is equal to
  26. From the base of a pole of height $20$ meter, the angle of elevation of the top of a tower is $60^{\circ}$. The pole subtends an angle $30^{\circ}$ at the top of the …
  27. The value of …
  28. The value of $\cot\frac{\pi}{24}$ is :
  29. $\alpha=\sin36^{\circ}$ is a root of which of the following equation?
  30. If for $\theta\in\left[-\frac{\pi}{3},0\right]$, the points $(x,y)=\left(3\tan\left(\theta+\frac{\pi}{3}\right),2\tan\left(\theta+\frac{\pi}{6}\right)\right)$ lie on …
  31. If $\alpha$, $-\frac{\pi}{2}<\alpha<\frac{\pi}{2}$ is the solution of $4\cos\theta+5\sin\theta=1$, then the value of $\tan\alpha$ is
  32. A spherical gas balloon of radius $16$ meter subtends an angle $60^{\circ}$ at the eye of the observer $A$ while the angle of elevation of its center from the eye of $A$ …
  33. The number of solutions of $|\cos x|=\sin x$, such that $-4\pi\le x\le4\pi$ is :
  34. In a triangle $ABC$, if $\cos A+2\cos B+\cos C=2$ and the lengths of the sides opposite to the angles $A$ and $C$ are $3$ and $7$ respectively, then $\cos A-\cos C$ is …
  35. The sum of all values of $x$ in $[0,2\pi]$, for which $\sin x+\sin2x+\sin3x+\sin4x=0$, is equal to :
  36. Let $S=\{\theta\in[0,2\pi]:8^{2\sin^2\theta}+8^{2\cos^2\theta}=16\}$. Then …
  37. $\operatorname{cosec}18^{\circ}$ is a root of the equation :
  38. The number of solutions of the equation $4\sin^{2}x-4\cos^{3}x+9-4\cos x=0$; $x\in[-2\pi,2\pi]$ is :
  39. The angle of elevation of the top $P$ of a tower from the feet of one person standing due South of the tower is $45^{\circ}$ and from the feet of another person standing …
  40. If $\cot\alpha=1$ and $\sec\beta=-\frac{5}{3}$, where $\pi<\alpha<\frac{3\pi}{2}$ and $\frac{\pi}{2}<\beta<\pi$, then the value of $\tan(\alpha+\beta)$ and the quadrant …
  41. Two poles, $AB$ of length $a$ metres and $CD$ of length $a+b$ ($b\neq a$) metres are erected at the same horizontal level with bases at $B$ and $D$. If $BD=x$ and …
  42. A tower $PQ$ stands on a horizontal ground with base $Q$ on the ground. The point $R$ divides the tower in two parts such that $QR=15$ m. If from a point $A$ on the …
  43. The sum of solutions of the equation $\frac{\cos x}{1+\sin x}=|\tan2x|$, $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)-\left\{\frac{\pi}{4},-\frac{\pi}{4}\right\}$ is :
  44. For $\alpha,\beta\in(0,\pi/2)$, let $3\sin(\alpha+\beta)=2\sin(\alpha-\beta)$ and a real number $k$ be such that $\tan\alpha=k\tan\beta$. Then, the value of $k$ is equal …
  45. The number of solutions of the equation $32^{\tan^{2}x}+32^{\sec^{2}x}=81$, $0\le x\le\frac{\pi}{4}$ is :
  46. The number of solutions of the equation $\left(4-\sqrt{3}\right)\sin x-2\sqrt{3}\cos^{2}x=-\frac{4}{1+\sqrt{3}}$, $x\in\left[-2\pi,\frac{5\pi}{2}\right]$ is
  47. If $10\sin^{4}\theta+15\cos^{4}\theta=6$, then the value of $\frac{27\operatorname{cosec}^{6}\theta+8\sec^{6}\theta}{16\sec^{8}\theta}$ is
  48. The value of $2\sin(12^{\circ})-\sin(72^{\circ})$ is :
  49. $2\sin\left(\frac\pi{22}\right)\sin\left(\frac{3\pi}{22}\right)\sin\left(\frac{5\pi}{22}\right)\sin\left(\frac{7\pi}{22}\right)\sin\left(\frac{9\pi}{22}\right)$ is equal …
  50. Let $a,b$ and $c$ be the length of sides of a triangle $ABC$ such that $\frac{a+b}{7}=\frac{b+c}{8}=\frac{c+a}{9}$. If $r$ and $R$ are the radius of incircle and radius …

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