Trigonometric Functions & Equations

Consider the curve $C_1$ given by
\(y=e^{-x} \quad \text { for } x \in[0,10 \pi],\)
and the curve $C_2$ given by
\(y=e^{-x}(\sin x+\cos x) \quad \text { for } x \in[0,10 \pi] .\)
Let $n$ be the total number of points of intersection of ...

Match each entry in List-I to the correct entry in List-II and choose the correct option.



List-I
List-II





(P) The number of elements in the set
$$\left\{x \in [-\pi,\pi] : \sin^6 ...

Let
\(\alpha=\frac{1}{\sin 60^{\circ} \sin 61^{\circ}}+\frac{1}{\sin 62^{\circ} \sin 63^{\circ}}+\cdots+\frac{1}{\sin 118^{\circ} \sin 119^{\circ}}\)
Then the value of
\(\left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2\)
is ...

Let $\frac{\pi}{2} < x < \pi$ be such that $\cot x=\frac{-5}{\sqrt{11}}$. Then

\(\left(\sin \frac{11 x}{2}\right)(\sin 6 x-\cos 6 x)+\left(\cos \frac{11 x}{2}\right)(\sin 6 x+\cos 6 x)\)

is equal to :

\(\text { Then the inradius of the triangle } A B C \text { is }\) :

Let $P Q R S$ be a quadrilateral in a plane, where $Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}$. If $\angle R P S=\theta^{\circ}, P Q=\alpha$ and $P S=\beta$, then the interval(s) that...

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi}{4}<\beta<0<\alpha<\frac{\pi}{4}$. If $\sin (\alpha+\beta)=\frac{1}{3}$ and $\cos (\alpha-\beta)=\frac{2}{3}$, then the greatest integer less than or equal to

$$
\left(\frac{\s...

Consider the following lists :

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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 > 0. Define the following sets whose elements are written in the increasing order :X = {x : f(x) = 0}, Y = {x : f'(x) = 0}Z = {x : g(x) = 0}, W = ...

Let f(x) = sin(\(\pi\) cos x) and g(x) = cos(2\(\pi\) sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:X = {x : f(x) = 0}, Y = {x : f'(x) = 0}Z = {x : g(x) = 0}, W = {...

For non-negative integers n, let$$f(n) = {{\sum\limits_{k = 0}^n {\sin \left( {{{k + 1} \over {n + 2}}\pi } \right)} \sin \left( {{{k + 2} \over {n + 2}}\pi } \right)} \over {\sum\limits_{k = 0}^n {{{\sin }^2}\left( {{{k + 1} \over {n + 2}}...

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'(x) f(y) for all x, y\(\in\) R.Then, the value of loge(f(4)) is ...........

In a \(\Delta\)PQR = 30\(^\circ\) and the sides PQ and QR have lengths 10\(\sqrt 3\) and 10, respectively. Then, which of the following statement(s) is(are) TRUE?

Let \(\alpha\) and \(\beta\) be non zero real numbers such that \(2(\cos \beta - \cos \alpha ) + \cos \alpha \cos \beta = 1\). Then which of the following is/are true?

If the triangle PQR varies, then the minimum value of cos(P + Q) + cos(Q + R) + cos(R + P) is

The value of \(\sum\limits_{k = 1}^{13} {{1 \over {\sin \left( {{\pi \over 4} + {{\left( {k - 1} \right)\pi } \over 6}} \right)\sin \left( {{\pi \over 4} + {{k\pi } \over 6}} \right)}}}\) is equal to

Let \(S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm {\pi \over 2}} \right\}.\) The sum of all distinct solutions of the equation \(\sqrt 3 \,\sec x + \cos ec\,x + 2\left( {\tan x - \cot x} \right) = 0\) in the set S is equa...

The number of distinct solutions of the equation
\({5 \over 4}{\cos ^2}\,2x + {\cos ^4}\,x + {\sin ^4}\,x + {\cos ^6}\,x + {\sin ^6}\,x\, = \,2\) in the interval \(\left[ {0,\,2\pi } \right]\) is

For \(x \in \left( {0,\pi } \right),\) the equation \(\sin x + 2\sin 2x - \sin 3x = 3\) has

The number of points in \(\left( { - \infty \,\infty } \right),\) for which \({x^2} - x\sin x - \cos x = 0,\) is

Let \(f\left( x \right) = x\sin \,\pi x,\,x > 0.\) Then for all natural numbers \(n,\,f'\left( x \right)\) vanishes at

Let \(\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]\) be such that
$$2\cos \theta \left( {1 - \sin \,\varphi } \right) = {\sin ^2}\theta \,\,\left( {\tan {\theta \over 2} + \cot {\theta \over 2}} \right)\cos \varphi - 1,\,\tan \le...

The positive integer value of \(n\, > \,3\) satisfying the equation \({1 \over {\sin \left( {{\pi \over n}} \right)}} = {1 \over {\sin \left( {{{2\pi } \over n}} \right)}} + {1 \over {\sin \left( {{{3\pi } \over n}} \right)}}\) is

Let \(P = \{ \theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta \}\) and \(Q = \{ \theta :\sin \theta + \cos \theta = \sqrt 2 \sin \theta \}\) be two sets. Then

Two parallel chords of a circle of radius 2 are at a distance \(\sqrt 3 + 1\) apart. If the chords subtend at the center , angles of \({\pi \over k}\) and \({{2\pi } \over k},\) where\(k > 0,\) then the value of \(\left[ k \right]\) is
[N...

The number of all possible values of \(\theta\) where \(0 < \theta < \pi ,\) for which the system of equations
\($\left( {y + z} \right)\cos {\mkern 1mu} 3\theta = \left( {xyz} \right){\mkern 1mu} \sin 3\theta\)$
$$$x\sin 3\theta = {{2...

The number of values of \(\theta\) in the interval, \(\left( { - {\pi \over 2},\,{\pi \over 2}} \right)\) such
that\(\,\theta \ne {{n\pi } \over 5}\) for \(n = 0,\, \pm 1,\, \pm 2\) and \(\tan \,\theta = \cot \,5\theta \,\) as well a...

The maximum value of the expression \({1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}\) is

For \(0 < \theta < {\pi \over 2},\) the solution (s) of
\($\sum\limits_{m = 1}^6 {\cos ec\,\left( {\theta + {{\left( {m - 1} \right)\pi } \over 4}} \right)\,\cos ec\,\left( {\theta + {{m\pi } \over 4}} \right) = 4\sqrt 2 }\)$ is (are)...

Match the statements/expressions in Column I with the values given in Column II:

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...

If \({{{{\sin }^4}x} \over 2} + {{{{\cos }^4}x} \over 3} = {1 \over 5},\) then

Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

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The number of solutions of the pair of equations
\(2{\sin ^2}\theta - \cos 2\theta = 0\)
\(2{\cos ^2}\theta - 3\sin \theta = 0\)
in the interval \([0,2\pi]\) is

The number of solutions of the pair of equations
\($\,2{\sin ^2}\theta - \cos 2\theta = 0\)$
\($2co{s^2}\theta - 3\sin \theta = 0\)$
in the interval \(\left[ {0,2\pi } \right]\)

If $0<\theta<2 \pi$, then the intervals of values of $\theta$ for which $2 \sin ^2 \theta-5 \sin \theta+2>0$, is

Let \(\theta \in\left(0, \frac{\pi}{4}\right)\) and \(t_{1}=(\tan \theta)^{\tan \theta}, t_{2}=(\tan \theta)^{\cot \theta}, t_{3}=(\cot \theta)^{\tan \theta}\) and \(t_{4}=(\cot \theta)^{\cot \theta}\), then

\(\cos \left( {\alpha - \beta } \right) = 1\) and \(\,\cos \left( {\alpha + \beta } \right) = 1/e\) where \(\alpha ,\,\beta \in \left[ { - \pi ,\pi } \right].\)
Paris of \(\alpha ,\,\beta\) which satisfy both the equations is/are

Find the range of values of \(\,t\) for which \($2\,\sin \,t = {{1 - 2x + 5{x^2}} \over {3{x^2} - 2x - 1}},\,\,\,\,\,t\, \in \,\left[ { - {\pi \over 2},\,{\pi \over 2}} \right].\)$

Given both \(\theta\) and \(\phi\) are acute angles and \(\sin \,\theta = {1 \over 2},\,\) \(\cos \,\phi = {1 \over 3},\) then the value of \(\theta + \phi\) belongs to

The number of integral values of \(k\) for which the equation \(7\cos x + 5\sin x = 2k + 1\) has a solution is

The maximum value of \(\left( {\cos {\alpha _1}} \right).\left( {\cos {\alpha _2}} \right).....\left( {\cos {\alpha _n}} \right),\) under the restrictions \(0 \le {\alpha _1},{\alpha _2},....,{\alpha _n} \le {\pi \over 2}\) vand $$\left( ...

If \(\alpha + \beta = \pi /2\) and \(\beta + \gamma = \alpha ,\) then \(\tan \,\alpha \,\) equals

The number of distinct real roots of \(\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right|\,\)
\(\, = 0\) in the interval $$ - {\pi \over ...

Let \(f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right)\). Then \(f\left( \theta \right)\) is

In any triangle \(ABC,\) prove that
\($\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.\)$

For a positive integer \(\,n\), let
$${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\left( {1 + \sec 2\theta } \right)\,\left( {1 + \sec 4\theta } \right).....\left( {1 + \sec {...

In a triangle \(PQR,\angle R = \pi /2\). If \(\,\,\tan \left( {P/2} \right)\) and \(\tan \left( {Q/2} \right)\) are the roots of the equation \(a{x^2} + bx + c = 0\left( {a \ne 0} \right)\) then.

Prove that \(\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha\)

The number of values of \(x\,\,\) in the interval \(\left[ {0,\,5\pi } \right]\) satisfying the equation \(3\,{\sin ^2}x - 7\,\sin \,x + 2 = 0\) is

Which of the following number(s) is /are rational?