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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 51-100 of 381

  1. If the value of $\frac{3\cos36^{\circ}+5\sin18^{\circ}}{5\cos36^{\circ}-3\sin18^{\circ}}$ is $\frac{a\sqrt{5}-b}{c}$, where $a,b,c$ are natural numbers and …
  2. A vertical pole consists of two portions, the lower being $\frac{1}{3}$rd of the whole. If the upper portion subtends an angle $\tan^{-1}\left(\frac{1}{2}\right)$ at a …
  3. The value of $36\left(4\cos^{2}9^{\circ}-1\right)\left(4\cos^{2}27^{\circ}-1\right)\left(4\cos^{2}81^{\circ}-1\right)\left(4\cos^{2}243^{\circ}-1\right)$ is
  4. The value of …
  5. Let $S=\left\{\theta\in[-\pi,\pi]-\left\{\pm\frac{\pi}{2}\right\}:\sin\theta\tan\theta+\tan\theta=\sin2\theta\right\}$. If $T=\sum_{\theta\in S}\cos2\theta$, then …
  6. Ten vertical poles of different height standing at equal distances on a straight line, subtend the same angle $30^{\circ}$ at a point $O$ on the line. If the height of …
  7. From the top $A$ of a vertical wall $AB$ of height $30$ m, the angles of depression of the top $P$ and bottom $Q$ of a vertical tower $PQ$ are $15^{\circ}$ and …
  8. If $\sin x=-\frac{3}{5}$, where $\pi<x<\frac{3\pi}{2}$, then $80\left(\tan^{2}x-\cos x\right)$ is equal to
  9. Let $|\cos\theta\cos(60-\theta)\cos(60+\theta)|\le\frac{1}{8}$, $\theta\in[0,2\pi]$. Then, the sum of all $\theta\in[0,2\pi]$, where $\cos3\theta$ attains its maximum …
  10. $16\sin(20^{\circ})\sin(40^{\circ})\sin(80^{\circ})$ is equal to :
  11. A $10$ inches long pencil $AB$ with mid point $C$ and a small eraser $P$ are placed on the horizontal top of a table such that $PC=\sqrt{5}$ inches and …
  12. The number of values of $x$ in the interval $\left(\frac{\pi}{4},\frac{7\pi}{4}\right)$ for which $14\operatorname{cosec}^{2}x-2\sin^{2}x=21-4\cos^{2}x$ holds, is …
  13. If the sum of solutions of the system of equations $2\sin^2\theta-\cos2\theta=0$ and $2\cos^2\theta+3\sin\theta=0$ in the interval $[0,2\pi]$ is $k\pi$, then $k$ is …
  14. If $\frac{\sqrt2\sin\alpha}{\sqrt{1+\cos2\alpha}}=\frac17$ and $\sqrt{\frac{1-\cos2\beta}{2}}=\frac1{\sqrt{10}}$, $\alpha,\beta\in\left(0,\frac\pi2\right)$, then …
  15. Let $S=\left[-\pi,\frac\pi2\right)-\left\{-\frac\pi2,-\frac\pi4,-\frac{3\pi}{4},\frac\pi4\right\}$. Then the number of elements in the set …
  16. In $\triangle ABC$, the lengths of sides $AC$ and $AB$ are $12$ cm and $5$ cm, respectively. If the area of $\triangle ABC$ is $30$ cm$^{2}$ and $R$ and $r$ are …
  17. The number of distinct solutions of the equation, $\log_{\frac12}|\sin x|=2-\log_{\frac12}|\cos x|$ in the interval $[0,2\pi]$, is ________.
  18. Let $S_1=\{x\in[0,12\pi]:\sin^{5}x+\cos^{5}x=1\}$ and $S_2=\{x\in[0,8\pi]:\sin^{7}x+\cos^{7}x=1\}$. Then $n(S_1)-n(S_2)$ is equal to $\_\_\_\_\_\_$ .
  19. If $\tan\left(\frac\pi9\right),x,\tan\left(\frac{7\pi}{18}\right)$ are in arithmetic progression and $\tan\left(\frac\pi9\right),y,\tan\left(\frac{5\pi}{18}\right)$ are …
  20. If the line $x=y=z$ intersects the line $x\sin A+y\sin B+z\sin C-18=0=x\sin2A+y\sin2B+z\sin2C-9$, where $A,B,C$ are the angles of a triangle $ABC$, then …
  21. The number of solutions of the equation $x^{2}=\cot x$ in $[0,2\pi]$ is $\_\_\_\_\_\_$ .
  22. The number of integral values of '$k$' for which the equation $3\sin x+4\cos x=k+1$ has a solution, $k\in\mathbf{R}$ is $\_\_\_\_\_\_$ .
  23. If the equation $\cos^4\theta+\sin^4\theta+\lambda=0$ has real solutions for $\theta$, then $\lambda$ lies in the interval :
  24. If $\sqrt{3}\left(\cos^{2}x\right)=(\sqrt{3}-1)\cos x+1$, the number of solutions of the given equation when $x\in\left[0,\frac{\pi}{2}\right]$ is $\_\_\_\_\_\_$ .
  25. The number of solutions of the equation $\sin x=\cos^{2}x$ in the interval $(0,10)$ is $\_\_\_\_\_\_$ .
  26. A horizontal park is in the shape of a triangle $OAB$ with $AB=16$. A vertical lamp post $OP$ is erected at the point $O$ such that $\angle PAO=\angle PBO=15^\circ$ and …
  27. The number of solutions of the equation $|\cot x|=\cot x+\frac{1}{\sin x}$ in the interval $[0,2\pi]$ is $\_\_\_\_\_\_$ .
  28. The angle of elevation of a cloud $C$ from a point $P$, $200$ m above a still lake is $30^\circ$. If the angle of depression of the image of $C$ in the lake from the …
  29. The number of solutions of the equation $2\theta-\cos^{2}\theta+\sqrt{2}=0$ in $\mathbf{R}$ is equal to $\_\_\_\_\_\_$ .
  30. If $S=\left\{\theta\in[-\pi,\pi]:\cos\theta\cos\frac{5\theta}{2}=\cos7\theta\cos\frac{7\theta}{2}\right\}$, then $n(S)$ is equal to $\_\_\_\_\_\_$ .
  31. Let $S$ be the sum of all solutions (in radians) of the equation $\sin^{4}\theta+\cos^{4}\theta-\sin\theta\cos\theta=0$ in $[0,4\pi]$. Then $\frac{8S}{\pi}$ is equal to …
  32. If $\sin^{2}(10^{\circ})\sin(20^{\circ})\sin(40^{\circ})\sin(50^{\circ})\sin(70^{\circ})=\alpha-\frac{1}{16}\sin(10^{\circ})$, then $16+\alpha^{-1}$ is equal to …
  33. The angle of elevation of the top of a tower from a point $A$ due north of it is $\alpha$ and from a point $B$ at a distance of $9$ units due west of $A$ is …
  34. Let the set of all $a\in\mathbf{R}$ such that the equation $\cos2x+a\sin x=2a-7$ has a solution be $[p,q]$ and …
  35. In the figure, $\theta_1+\theta_2=\frac{\pi}{2}$ and $\sqrt{3}(BE)=4(AB)$. If the area of $\triangle CAB$ is $2\sqrt{3}-3$ unit$^{2}$, when $\frac{\theta_2}{\theta_1}$ …
  36. The value of $\cos^3\left(\frac\pi8\right)\cdot\cos\left(\frac{3\pi}8\right)+\sin^3\left(\frac\pi8\right)\cdot\sin\left(\frac{3\pi}8\right)$ is :
  37. The value of $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$ is $\_\_\_\_\_\_$ .
  38. If $\sin\theta+\cos\theta=\frac12$, then $16(\sin(2\theta)+\cos(4\theta)+\sin(6\theta))$ is equal to :
  39. In a triangle $ABC$, $BC=7$, $AC=8$, $AB=\alpha\in\mathbf{N}$ and $\cos A=\frac{2}{3}$. If $49\cos(3C)+42=\frac{m}{n}$, where $\gcd(m,n)=1$, then $m+n$ is equal to …
  40. Two vertical poles $AB=15$ m and $CD=10$ m are standing apart on a horizontal ground with points $A$ and $C$ on the ground. If $P$ is the point of intersection of $BC$ …
  41. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x+y)+f(x-y)=2f(x)f(y)$, $f\left(\frac12\right)=-1$. Then, the value of $\sum_{k=1}^{20}\frac{1}{\sin(k)\sin(k+f(k))}$ is …
  42. Let $S=\{\theta\in(0,2\pi):7\cos^2\theta-3\sin^2\theta-2\cos^22\theta=2\}$. Then, the sum of roots of all the equations …
  43. The number of elements in the set $S=\left\{x\in\mathbb{R}:2\cos\left(\frac{x^2+x}{6}\right)=4^x+4^{-x}\right\}$ is :
  44. Let a vertical tower $AB$ of height $2h$ stand on a horizontal ground. Let from a point $P$ on the ground a man can see upto height $h$ of the tower with an angle of …
  45. Let $S=\left\{\theta\in\left(0,\frac\pi2\right):\sum_{m=1}^{9}\sec\left(\theta+(m-1)\frac\pi6\right)\sec\left(\theta+\frac{m\pi}{6}\right)=-\frac{8}{\sqrt3}\right\}$. …
  46. The angle of elevation of the top $P$ of a vertical tower $PQ$ of height $10$ from a point $A$ on the horizontal ground is $45^\circ$. Let $R$ be a point on $AQ$ and …
  47. The number of elements in the set $S=\left\{\theta\in[-4\pi,4\pi]:3\cos^{2}2\theta+6\cos2\theta-10\cos^{2}\theta+5=0\right\}$ is $\_\_\_\_\_\_$ .
  48. If $x=\sum_{n=0}^{\infty}(-1)^n\tan^{2n}\theta$ and $y=\sum_{n=0}^{\infty}\cos^{2n}\theta$, for $0<\theta<\frac\pi4$, then :
  49. Let $\tan\alpha$, $\tan\beta$ and $\tan\gamma$; $\alpha,\beta,\gamma\neq\frac{(2n-1)\pi}{2}$, $n\in\mathbf{N}$ be the slopes of three line segments $OA$, $OB$ and $OC$ …
  50. The angle of elevation of the top $P$ of a vertical tower for a person standing at a point $A$ on the horizontal ground due north of the tower is $45^\circ$. Another …

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