Trigonometric Functions & Equations

Prove that the values of the function \({{\sin x\cos 3x} \over {\sin 3x\cos x}}\) do not lie between \({1 \over 3}\) and 3 for any real \(x.\)

Prove that \(\sum\limits_{k = 1}^{n - 1} {\left( {n - k} \right)\,\cos \,{{2k\pi } \over n} = - {n \over 2},}\) where \(n \ge 3\) is an integer.

The real roots of the equation \(\,{\cos ^7}x + {\sin ^4}x = 1\) in the interval \(\left( { - \pi ,\pi } \right)\) are ...., ...., and ______.

\({\sec ^2}\theta = {{4xy} \over {{{\left( {x + y} \right)}^2}}}\,\) is true if and only if

Find all values of \(\theta\) in the interval \(\left( { - {\pi \over 2},{\pi \over 2}} \right)\) satisfying the equation $$\left( {1 - \tan \,\theta } \right)\left( {1 + \tan \,\theta } \right)\,\,{\sec ^2}\theta + \,\,{2^{{{\tan }^2}...

General value of \(\theta\) satisfying the equation \({\tan ^2}\theta + \sec \,2\,\theta = 1\) is _________.

The general values of \(\theta\) satisfying the equation \(2{\sin ^2}\theta - 3\sin \theta - 2 = 0\) is

\(\,3{\left( {\sin x - \cos x} \right)^4} + 6{\left( {\sin x + \cos x} \right)^2} + 4\left( {{{\sin }^6}x + {{\cos }^6}x} \right) =\)

The minimum value of the expression \(\sin \,\alpha + \sin \,\beta \, + \sin \,\gamma ,\,\) where \(\alpha ,\,\beta ,\,\gamma\) are real numbers satisfying \(\alpha + \beta + \gamma = \pi\) is

Find the smallest positive number \(p\) for which the equation \(\cos \left( {p\,\sin x} \right) = \sin \left( {p\cos x} \right)\) has a solution \(x\, \in \,\left[ {0,2\pi } \right]\).

Let \(2{\sin ^2}x + 3\sin x - 2 > 0\) and \({x^2} - x - 2 < 0\) (\(x\) is measured in radians). Then \(x\) lies in the interval

If \(\omega \,\) is an imaginary cube root of unity then the value of \(\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - {\pi \over 4}} \right\}\) is

Let \(0 < x < {\pi \over 4}\) then \(\left( {\sec 2x - \tan 2x} \right)\) equals

Let \(n\) be a positive integer such that \(\sin {\pi \over {2n}} + \cos {\pi \over {2n}} = {{\sqrt n } \over 2}.\) Then

If \(K = \sin \left( {\pi /18} \right)\sin \left( {5\pi /18} \right)\sin \left( {7\pi /18} \right),\) then the numerical value of K is ______.

Determine the smallest positive value of number \(x\) (in degrees) for which
\($\tan \left( {x + {{100}^ \circ }} \right) = \tan \left( {x + {{50}^ \circ }} \right)\,\tan \left( x \right)\tan \left( {x - {{50}^ \circ }} \right).\)$

If \(A > 0,B > 0\,\) and \(A + B = \pi /3,\) then the maximum value of tan A tan B is _______.

Number of solutions of the equation \(\tan x + \sec x = 2\cos x\,\) lying in the interval \(\left[ {0,2\pi } \right]\) is:

Show that the value of \({{\tan x} \over {\tan 3x}},\) wherever defined never lies between \({1 \over 3}\) and 3.

In this questions there are entries in columns 1 and 2. Each entry in column 1 is related to exactly one entry in column 2. Write the correct letter from column 2 against the entry number in column 1 in your answer book.
$${{\sin \,3\alpha ...

The value of
\(\sin {\pi \over {14}}\sin {{3\pi } \over {14}}\sin {{5\pi } \over {14}}\sin {{7\pi } \over {14}}\sin {{9\pi } \over {14}}\sin {{11\pi } \over {14}}\sin {{13\pi } \over {14}}\) is equal to ______.

If \(\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \right)\,In\,\,2} \right)} \right\}\) satiesfies the equation \({x^2} - 9x + 8 = 0,\) find the value of $${{\cos x} \over {\...

The equation \(\left( {\cos p - 1} \right){x^2} + \left( {\cos p} \right)x + \sin p = 0\,\)
In the variable x, has real roots. Then p can take any value in the interval

\(ABC\) is a triangle such that
\($\sin \left( {2A + B} \right) = \sin \left( {C - A} \right) = \, - \sin \left( {B + 2C} \right) = {1 \over 2}.\)$
If \(A,\,B\) and \(C\) are in arithmetic progression, determine the values of \(A,\,B\) and...

The general solutions of \(\,\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x\) is

The value of the expression \(\sqrt 3 \,\cos \,ec\,{20^0} - \sec \,{20^0}\) is equal to

The values of \(\theta\) lying between \(\theta = \theta\) and \(\theta = \pi /2\) and satisfying the equation
$$\left| {\matrix{
{1 + {{\sin }^2}\theta } & {{{\cos }^2}\theta } & {4\sin 4\theta } \cr
{{{\sin }^2}\theta } & {1 ...

The number of all possible triplets \(\left( {{a_1},\,{a_2},\,{a_3}} \right)\) such that \({a_1} + {a_2}\,\,\cos \left( {2x} \right) + {a_3}{\sin ^2}\left( x \right) = 0\,\) for all \(x\) is

The set of all \(x\) in the interval \(\left[ {0,\,\pi } \right]\) for which \(2\,{\sin ^2}x - 3\) \(\sin x + 1 \ge 0,\) is _____.

The solution set of the system of equations \(X + Y = {{2\pi } \over 3},\) \(cox\,x + cos\,y = {3 \over 2},\) where x and y are real, is _____.

The sides of a triangle inscribed in a given circle subtend angles \(\alpha\), \(\beta\) and \(\gamma\) at the centre. The minimum value of the arithmetic mean of $$cos\left[ {\alpha + {\pi \over 2}} \right],\,\cos \left[ {\beta + {\p...

The expression \(2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]\) is equal to

\(\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}} \right)\left( {1 + \cos {{7\pi } \over 8}} \right)\) is equal to

There exists a value of \(\theta\) between 0 and \(2\pi\) that satisfies the equation \(\,\,{\sin ^4}\theta - 2{\sin ^2}\theta - 1 = 0.\)

Find the values of \(x \in \left( { - \pi , + \pi } \right)\) which satisfy the equation \({g^{(1 + \left| {\cos x} \right| + \left| {{{\cos }^2}x} \right| + \left| {{{\cos }^3}x} \right| + ...)}} = {4^3}\)

If \(\tan \,A = \left( {1 - \cos B} \right)/\sin B,\) then \(tan2A = tan\,B\).

Find all solutions of \(4{\cos ^2}\,x\sin x - 2{\sin ^2}x = 3\sin x\)

Show that \($16\cos \left( {{{2\pi } \over {15}}} \right)\cos \left( {{{4\pi } \over {15}}} \right)\cos \left( {{{8\pi } \over {15}}} \right)\cos \left( {{{16\pi } \over {15}}} \right) = 1\)$

Without using tables, prove that \(\left( {\sin \,{{12}^ \circ }} \right)\left( {\sin \,{{48}^ \circ }} \right)\left( {\sin \,{{54}^ \circ }} \right) = {1 \over 8}.\)

The general solution of the trigonometric equation sin x+cos x=1 is given by:

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 constants, and \({C_n} \ne 0\) , then the value of n is _____.

For all \(\theta\) in \(\left[ {0,\,\pi /2} \right]\) show that, \(\cos \left( {\sin \theta } \right) \ge \,\sin \,\left( {\cos \theta } \right).\)

Given \(A = \left\{ {x:{\pi \over 6} \le x \le {\pi \over 3}} \right\}\) and
\(f\left( x \right) = \cos x - x\left( {1 + x} \right);\) find \(f\left( A \right).\)

Given \(A = {\sin ^2}\theta + {\cos ^4}\theta\) then for all real values of \(\theta\)

The equation \(\,2{\cos ^2}{x \over 2}{\sin ^2}x = {x^2} + {x^{ - 2}};\,0 < x \le {\pi \over 2}\) has

Given \(\alpha + \beta - \gamma = \pi ,\) prove that
\(\,{\sin ^2}\alpha + {\sin ^2}\beta - {\sin ^2}\gamma = 2\sin \alpha {\mkern 1mu} \sin \beta {\mkern 1mu} \cos y\)

(a) Draw the graph of \(y = {1 \over {\sqrt 2 }}\left( {cinx + \cos x} \right)\) from \(x = - {\pi \over 2}\) to \(x = {\pi \over 2}\).
(b) If $$\cos \left( {\alpha + \beta } \right) = {4 \over 5},\,\,\sin \,\left( {\alpha - \beta } \r...

If \(\tan \theta = - {4 \over 3},then\sin \theta \,is\,\)

If \(\alpha + \beta + \gamma = 2\pi ,\) then

If \(\tan \alpha = {m \over {m + 1}}\,\) and \(\tan \beta = {2 \over {2m + 1}},\) find the possible values of \(\left( {\alpha + \beta } \right).\)