36^ 72^ 108^ 144^ = - Vidyadip

MCQ

$\sin 36^\circ \sin 72^\circ \sin 108^\circ \sin 144^\circ = $

A
$1/4$
B
$1/16$
C
$3/4$
✓
$5/16$

✓

Answer

Correct option: D.

$5/16$

d
(d) $\sin \,\,{36^o}\,\,\sin \,\,{72^o}\,\sin \,\,{108^o}\,\,\sin \,\,{144^o}$

$ = {\sin ^2}{36^o}\,\,{\sin ^2}\,{72^o} = \frac{1}{4}\,\left\{ {(2\,\,{{\sin }^2}{{36}^o})\,\,(2\,\,{{\sin }^2}\,\,{{72}^o})} \right\}$

$ = \frac{1}{4}\left\{ {(1 - \cos \,\,{{72}^o})\,\,(1 - \cos \,\,{{144}^o})} \right\}$

$ = \frac{1}{4}\left\{ {(1 - \sin \,\,{{18}^o})\,\,(1 + \cos \,\,{{36}^o})} \right\}$

$ = \frac{1}{4}\left[ {\left( {1 - \frac{{\sqrt 5 - 1}}{4}} \right)\,\,\left( {1 + \frac{{\sqrt 5 + 1}}{4}} \right)} \right]$

$= \frac{{20}}{{16}} \times \frac{1}{4} = \frac{5}{{16}}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $1, \log _{10}\left(4^{x}-2\right)$ and $\log _{10}\left(4^{x}+\frac{18}{5}\right)$ are in
arithmetic progression for a real number $x$ then the value of the determinant $\left|\begin{array}{ccc}2\left(x-\frac{1}{2}\right) & x-1 & x^{2} \\ 1 & 0 & x \\ x & 1 & 0\end{array}\right|$ is equal to ...... .

The number of distinct real roots of $\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is

Let the mean and standard deviation of marks of class $A$ of $100$ students be respectively $40$ and $\alpha( > 0)$, and the mean and standard deviation of marks of class $B$ of $n$ students be respectively $55$ and $30-\alpha$. If the mean and variance of the marks of the combined class of $100+ n$ students are respectively $50$ and $350$,then the sum of variances of classes $A$ and $B$ is

If ${2^x} + {2^y} = {2^{x + y}}$, then ${{dy} \over {dx}} = $

If $g(f(x)) = |\sin x|$ and $f(g(x)) = {(\sin \sqrt x )^2}$, then

If $\frac{\pi }{2} < \alpha < \frac{3}{2}\pi $ , then the modulus and argument of $(1 + cos\, 2\alpha ) + i\, sin\, 2\alpha $ is respectively

The function $f:(-\infty, \infty) \rightarrow(-\infty, 1)$, defined by $f(x)=\frac{2^{x}-2^{-x}}{2^{x}+2^{-x}}$ is :

Let $a$, $b \in R$ be such that $a$, $a + 2b$ , $2a + b$ are in $A.P$. and $(b + 1)^2$, $ab + 5$, $(a + 1)^2$ are in $G.P.$ then $(a + b)$ equals

If $\tan \theta + \tan 2\theta + \sqrt 3 \tan \theta \tan 2\theta = \sqrt 3 ,$ then

If $a, b$ and $c$ be three non-zero vectors, no two of which are collinear. If the vector $a + 2b$ is collinear with $ c$ and $b + 3c$ is collinear with a, then ($\lambda $ being some non-zero scalar) $a + 2b + 6c$ is equal to