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## Trigonometric Functions & Equations

### Mathematics

#### Previous Years Questions

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Let f(x) = sin($$\pi$$ cos x) and g(x) = cos(2$$\pi$$ sin x) be two functions defined for x &gt; 0. Define the followi...
JEE Advanced 2019 Paper 2 Offline
Let f(x) = sin($$\pi$$ cos x) and g(x) = cos(2$$\pi$$ sin x) be two functions defined for x &gt; 0. Define the followi...
JEE Advanced 2019 Paper 2 Offline
The value of $$\sum\limits_{k = 1}^{13} {{1 \over {\sin \left( {{\pi \over 4} + {{\left( {k - 1} \right)\pi } \over 6}}... JEE Advanced 2016 Paper 2 Offline Let$$S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm {\pi \over 2}} \right\}.$$The sum of all distinct ... JEE Advanced 2016 Paper 1 Offline For$$x \in \left( {0,\pi } \right),$$the equation$$\sin x + 2\sin 2x - \sin 3x = 3$$has JEE Advanced 2014 Paper 2 Offline The number of points in$$\left( { - \infty \,\infty } \right),$$for which$${x^2} - x\sin x - \cos x = 0,$$is JEE Advanced 2013 Paper 1 Offline Let$$P = \{ \theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta \} $$and$$Q = \{ \theta :\sin \theta + \cos \t...
IIT-JEE 2011 Paper 1 Offline
The number of solutions of the pair of equations $$\,2{\sin ^2}\theta - \cos 2\theta = 0$$$$$2co{s^2}\theta - 3\si... IIT-JEE 2007 Let$$\theta \in \left( {0,{\pi \over 4}} \right)$$and$${t_1} = {\left( {\tan \theta } \right)^{\tan \theta }},\,\,\... IIT-JEE 2006 The values of $$\theta \in \left( {0,2\pi } \right)$$ for which $$2\,{\sin ^2}\theta - 5\,\sin \theta + 2 &gt; 0,$$ a... IIT-JEE 2006 Screening $$\cos \left( {\alpha - \beta } \right) = 1$$ and $$\,\cos \left( {\alpha + \beta } \right) = 1/e$$ where $$\alpha ,\,... IIT-JEE 2005 Screening Given both$$\theta $$and$$\phi $$are acute angles and$$\sin \,\theta = {1 \over 2},\,\cos \,\phi = {1 \over ... IIT-JEE 2004 Screening The number of integral values of $$k$$ for which the equation $$7\cos x + 5\sin x = 2k + 1$$ has a solution is IIT-JEE 2002 Screening If $$\alpha + \beta = \pi /2$$ and $$\beta + \gamma = \alpha ,$$ then $$\tan \,\alpha \,$$ equals IIT-JEE 2001 Screening The maximum value of $$\left( {\cos {\alpha _1}} \right).\left( {\cos {\alpha _2}} \right).....\left( {\cos {\alpha _n}}... IIT-JEE 2001 Screening The number of distinct real roots of$$\left| {\matrix{ {\sin x} &amp; {\cos x} &amp; {\cos x} \cr {\cos x} &amp... IIT-JEE 2001 Screening Let $$f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right)$$. Then $$f\left( \theta \rig... IIT-JEE 2000 Screening In a triangle$$PQR,\angle R = \pi /2$$. If$$\,\,\tan \left( {P/2} \right)$$and$$\tan \left( {Q/2} \right)$$are the... IIT-JEE 1999 Which of the following number(s) is /are rational? IIT-JEE 1998 The number of values of$$x\,\,$$in the interval$$\left[ {0,\,5\pi } \right]$$satisfying the equation$$3\,{\sin ^2}x... IIT-JEE 1998 $${\sec ^2}\theta = {{4xy} \over {{{\left( {x + y} \right)}^2}}}\,$$ is true if and only if IIT-JEE 1996 The minimum value of the expression $$\sin \,\alpha + \sin \,\beta \, + \sin \,\gamma ,\,$$ where $$\alpha ,\,\beta ,\... IIT-JEE 1995 The general values of$$\theta $$satisfying the equation$$2{\sin ^2}\theta - 3\sin \theta - 2 = 0$$is IIT-JEE 1995 Screening$$\,3{\left( {\sin x - \cos x} \right)^4} + 6{\left( {\sin x + \cos x} \right)^2} + 4\left( {{{\sin }^6}x + {{\cos }^6}x... IIT-JEE 1995 Screening Let $$2{\sin ^2}x + 3\sin x - 2 &gt; 0$$ and $${x^2} - x - 2 &lt; 0$$ ($$x$$ is measured in radians). Then $$x$$ lies in... IIT-JEE 1994 If $$\omega \,$$ is an imaginary cube root of unity then the value of $$\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^... IIT-JEE 1994 Let$$n$$be a positive integer such that$$\sin {\pi \over {2n}} + \cos {\pi \over {2n}} = {{\sqrt n } \over 2}.$$T... IIT-JEE 1994 Let$$0 &lt; x &lt; {\pi \over 4}$$then$$\left( {\sec 2x - \tan 2x} \right)$$equals IIT-JEE 1994 Number of solutions of the equation$$\tan x + \sec x = 2\cos x\,$$lying in the interval$$\left[ {0,2\pi } \right]$$i... IIT-JEE 1993 In this questions there are entries in columns 1 and 2. Each entry in column 1 is related to exactly one entry in column... IIT-JEE 1992 The equation$$\left( {\cos p - 1} \right){x^2} + \left( {\cos p} \right)x + \sin p = 0\,$$In the variable x, has real... IIT-JEE 1990 The general solutions of$$\,\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x$$is IIT-JEE 1989 The value of the expression$$\sqrt 3 \,\cos \,ec\,{20^0} - \sec \,{20^0}$$is equal to IIT-JEE 1988 The number of all possible triplets$$\left( {{a_1},\,{a_2},\,{a_3}} \right)$$such that$${a_1} + {a_2}\,\,\cos \left( ... IIT-JEE 1987 The expression $$2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \ri... IIT-JEE 1986$$\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}}... IIT-JEE 1984 The general solution of the trigonometric equation sin x+cos x=1 is given by: IIT-JEE 1981 The equation $$\,2{\cos ^2}{x \over 2}{\sin ^2}x = {x^2} + {x^{ - 2}};\,0 &lt; x \le {\pi \over 2}$$ has IIT-JEE 1980 Given $$A = {\sin ^2}\theta + {\cos ^4}\theta$$ then for all real values of $$\theta$$ IIT-JEE 1980 If $$\alpha + \beta + \gamma = 2\pi ,$$ then IIT-JEE 1979 If $$\tan \theta = - {4 \over 3},then\sin \theta \,is\,$$ IIT-JEE 1979 ## MCQ (More than One Correct Answer) More For non-negative integers n, let$$f(n) = {{\sum\limits_{k = 0}^n {\sin \left( {{{k + 1} \over {n + 2}}\pi } \right)} \si... JEE Advanced 2019 Paper 2 Offline In a$$\Delta $$PQR = 30$$^\circ $$and the sides PQ and QR have lengths 10$$\sqrt 3 $$and 10, respectively. Then, whic... JEE Advanced 2018 Paper 1 Offline Let$$\alpha $$and$$\beta $$be non zero real numbers such that$$2(\cos \beta - \cos \alpha ) + \cos \alpha \cos \be... JEE Advanced 2017 Paper 2 Offline Let $$f\left( x \right) = x\sin \,\pi x,\,x &gt; 0.$$ Then for all natural numbers $$n,\,f'\left( x \right)$$ vanishes a... JEE Advanced 2013 Paper 1 Offline Let $$\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]$$ be such that $$2\cos \theta \left( {1 - \sin \,\varphi } \r... IIT-JEE 2012 Paper 1 Offline If$${{{{\sin }^4}x} \over 2} + {{{{\cos }^4}x} \over 3} = {1 \over 5},$$then IIT-JEE 2009 For$$0 &lt; \theta &lt; {\pi \over 2},$$the solution (s) of$$$\sum\limits_{m = 1}^6 {\cos ec\,\left( {\theta + {{...
IIT-JEE 2009
For a positive integer $$\,n$$, let $${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {... IIT-JEE 1999 The values of$$\theta $$lying between$$\theta = \theta $$and$$\theta = \pi /2$$and satisfying the equation$$\l...
IIT-JEE 1988

## Numerical

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Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the X-axis, Y-axis and Z-axis, respect...
JEE Advanced 2018 Paper 2 Offline
Let f : R $$\to$$ R be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x) f'(y) + f'(...
JEE Advanced 2018 Paper 2 Offline
The number of distinct solutions of the equation $${5 \over 4}{\cos ^2}\,2x + {\cos ^4}\,x + {\sin ^4}\,x + {\cos ^6}\... JEE Advanced 2015 Paper 1 Offline The positive integer value of$$n\, &gt; \,3$$satisfying the equation$${1 \over {\sin \left( {{\pi \over n}} \right)}...
IIT-JEE 2011 Paper 1 Offline
Two parallel chords of a circle of radius 2 are at a distance $$\sqrt 3 + 1$$ apart. If the chords subtend at the cente...
IIT-JEE 2010 Paper 2 Offline
The maximum value of the expression $${1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}$$...
IIT-JEE 2010 Paper 1 Offline
The number of values of $$\theta$$ in the interval, $$\left( { - {\pi \over 2},\,{\pi \over 2}} \right)$$ such that...
IIT-JEE 2010 Paper 1 Offline
The number of all possible values of $$\theta$$ where $$0 &lt; \theta &lt; \pi ,$$ for which the system of equations $... IIT-JEE 2010 Paper 1 Offline ## Subjective More Find the range of values of $$\,t$$ for which $$2\,\sin \,t = {{1 - 2x + 5{x^2}} \over {3{x^2} - 2x - 1}},\,\,\,\,\,t\,... IIT-JEE 2005 In any triangle$$ABC,$$prove that$$$\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B...
IIT-JEE 2000
Prove that $$\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha$$
IIT-JEE 1998
Prove that $$\sum\limits_{k = 1}^{n - 1} {\left( {n - k} \right)\,\cos \,{{2k\pi } \over n} = - {n \over 2},}$$ where ...
IIT-JEE 1997
Prove that the values of the function $${{\sin x\cos 3x} \over {\sin 3x\cos x}}$$ do not lie between $${1 \over 3}$$ and...
IIT-JEE 1997
Find all values of $$\theta$$ in the interval $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$ satisfying the equ...
IIT-JEE 1996
Find the smallest positive number $$p$$ for which the equation $$\cos \left( {p\,\sin x} \right) = \sin \left( {p\cos x}... IIT-JEE 1995 Determine the smallest positive value of number$$x$$(in degrees) for which$$$\tan \left( {x + {{100}^ \circ }} \rig... IIT-JEE 1993 Show that the value of $${{\tan x} \over {\tan 3x}},$$ wherever defined never lies between $${1 \over 3}$$ and 3. IIT-JEE 1992 If $$\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \righ... IIT-JEE 1991$$ABC$$is a triangle such that$$$\sin \left( {2A + B} \right) = \sin \left( {C - A} \right) = \, - \sin \left( {B + 2...
IIT-JEE 1990
Find the values of $$x \in \left( { - \pi , + \pi } \right)$$ which satisfy the equation $${g^{(1 + \left| {\cos x} \rig... IIT-JEE 1984 Show that$$$16\cos \left( {{{2\pi } \over {15}}} \right)\cos \left( {{{4\pi } \over {15}}} \right)\cos \left( {{{8\pi }... IIT-JEE 1983 Find all solutions of $$4{\cos ^2}\,x\sin x - 2{\sin ^2}x = 3\sin x$$ IIT-JEE 1983 Without using tables, prove that $$\left( {\sin \,{{12}^ \circ }} \right)\left( {\sin \,{{48}^ \circ }} \right)\left( {... IIT-JEE 1982 For all$$\theta $$in$$\left[ {0,\,\pi /2} \right]$$show that,$$\cos \left( {\sin \theta } \right) \ge \,\sin \,\le... IIT-JEE 1980 Given $$A = \left\{ {x:{\pi \over 6} \le x \le {\pi \over 3}} \right\}$$ and $$f\left( x \right) = \cos x - x\left( {... IIT-JEE 1980 Given$$\alpha + \beta - \gamma = \pi ,$$prove that$$\,{\sin ^2}\alpha + {\sin ^2}\beta - {\sin ^2}\gamma = 2\s... IIT-JEE 1980 (a) Draw the graph of $$y = {1 \over {\sqrt 2 }}\left( {cinx + \cos x} \right)$$ from $$x = - {\pi \over 2}$$ to $$x =... IIT-JEE 1979 If$$\tan \alpha = {m \over {m + 1}}\,$$and$$\tan \beta = {2 \over {2m + 1}},$$find the possible values of$$\left... IIT-JEE 1978 ## Fill in the Blanks More The real roots of the equation $$\,{\cos ^7}x + {\sin ^4}x = 1$$ in the interval $$\left( { - \pi ,\pi } \right)$$ are .... IIT-JEE 1997 General value of $$\theta$$ satisfying the equation $${\tan ^2}\theta + \sec \,2\,\theta = 1$$ is _________. IIT-JEE 1996 If $$A &gt; 0,B &gt; 0\,$$ and $$A + B = \pi /3,$$ then the maximum value of tan A tan B is _______. IIT-JEE 1993 If $$K = \sin \left( {\pi /18} \right)\sin \left( {5\pi /18} \right)\sin \left( {7\pi /18} \right),$$ then the numerica... IIT-JEE 1993 The value of $$\sin {\pi \over {14}}\sin {{3\pi } \over {14}}\sin {{5\pi } \over {14}}\sin {{7\pi } \over {14}}\sin {{... IIT-JEE 1991 The set of all$$x$$in the interval$$\left[ {0,\,\pi } \right]$$for which$$2\,{\sin ^2}x - 3\sin x + 1 \ge 0,$...
IIT-JEE 1987
The sides of a triangle inscribed in a given circle subtend angles $$\alpha$$, $$\beta$$ and $$\gamma$$ at the centre...
IIT-JEE 1987
The solution set of the system of equations $$X + Y = {{2\pi } \over 3},$$ $$cox\,x + cos\,y = {3 \over 2},$$ where x an...
IIT-JEE 1987
Suppose $${\sin ^3}\,x\sin 3x = \sum\limits_{m = 0}^n {{C_m}\cos \,mx}$$ is an identity in x, where C0, C1 ,....Cn are...
IIT-JEE 1981

## True or False

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There exists a value of $$\theta$$ between 0 and $$2\pi$$ that satisfies the equation $$\,\,{\sin ^4}\theta - 2{\sin ... IIT-JEE 1984 If$$\tan \,A = \left( {1 - \cos B} \right)/\sin B,$$then$$tan2A = tan\,B.
IIT-JEE 1983

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