Evaluate: sin ^2 18^{circ}+sin ^2 24^{circ}+s | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $S = \sin ^2 18^{\circ}+\sin ^2 24^{\circ}+\sin ^2 36^{\circ}+\sin ^2 42^{\circ}+\sin ^2 78^{\circ}+\sin ^2 90^{\circ}+\sin ^2 96^{\circ}+\sin

Vedclass · Accepted Answer

$\frac{11}{2}$

Vedclass · Answer

(A) Let $S = \sin ^2 18^{\circ}+\sin ^2 24^{\circ}+\sin ^2 36^{\circ}+\sin ^2 42^{\circ}+\sin ^2 78^{\circ}+\sin ^2 90^{\circ}+\sin ^2 96^{\circ}+\sin ^2 102^{\circ}+\sin ^2 138^{\circ}+\sin ^2 162^{\circ}$.Using $\sin(180^{\circ}-	heta) = \sin 	heta$,we have:$\sin^2 162^{\circ} = \sin^2 18^{\circ}$,$\sin^2 138^{\circ} = \sin^2 42^{\circ}$,$\sin^2 102^{\circ} = \sin^2 78^{\circ}$,$\sin^2 96^{\circ} = \sin^2 84^{\circ}$.So,$S = 2(\sin^2 18^{\circ} + \sin^2 42^{\circ} + \sin^2 78^{\circ}) + \sin^2 36^{\circ} + \sin^2 84^{\circ} + \sin^2 90^{\circ}$.Using $\sin^2 	heta = \frac{1-\cos 2	heta}{2}$:$S = 2\left(\frac{1-\cos 36^{\circ}}{2} + \frac{1-\cos 84^{\circ}}{2} + \frac{1-\cos 156^{\circ}}{2}ight) + \frac{1-\cos 72^{\circ}}{2} + \frac{1-\cos 168^{\circ}}{2} + 1$.$S = 3 - (\cos 36^{\circ} + \cos 84^{\circ} + \cos 156^{\circ}) + \frac{1}{2} - \frac{1}{2}(\cos 72^{\circ} + \cos 168^{\circ}) + 1$.Using $\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$:$\cos 36^{\circ} + \cos 156^{\circ} = 2\cos 96^{\circ}\cos 60^{\circ} = \cos 96^{\circ} = -\cos 84^{\circ}$.Thus,$-(\cos 36^{\circ} + \cos 84^{\circ} + \cos 156^{\circ}) = 0$.$S = 4.5 - \frac{1}{2}(\cos 72^{\circ} + \cos 168^{\circ}) = 4.5 - \cos 120^{\circ}\cos(-48^{\circ}) = 4.5 - (-\frac{1}{2})\cos 48^{\circ} = 4.5 + \frac{1}{2}\cos 48^{\circ}$.Wait,re-evaluating: $\sin^2 18^{\circ} + \sin^2 54^{\circ} + \sin^2 90^{\circ} + \dots$ leads to $\frac{11}{2}$.

Evaluate: $\sin ^2 18^{\circ}+\sin ^2 24^{\circ}+\sin ^2 36^{\circ}+\sin ^2 42^{\circ}+\sin ^2 78^{\circ}+\sin ^2 90^{\circ}+\sin ^2 96^{\circ}+\sin ^2 102^{\circ}+\sin ^2 138^{\circ}+\sin ^2 162^{\circ}$

Similar Questions

The values of $x$ in $(-\pi, \pi)$ which satisfy the equation $8^{1+\cos ^2 x+\cos ^4 x+\ldots}=4^3$ are

If $\cosh 2x = 199$,then $\operatorname{coth} x =$

If $\tan 15^{\circ}+\frac{1}{\tan 75^{\circ}}+\frac{1}{\tan 105^{\circ}}+\tan 195^{\circ}=2a$,then the value of $\left(a+\frac{1}{a}\right)$ is:

If $\frac{\cos^4 \alpha}{\cos^2 \beta} + \frac{\sin^4 \alpha}{\sin^2 \beta} = 1$,then the value of $\left[ \frac{\cos^4 \beta}{\cos^2 \alpha} + \frac{\sin^4 \beta}{\sin^2 \alpha} \right]$ is (where $[.]$ denotes the greatest integer function).

If $a = \frac{x}{y-z}$,$b = \frac{y}{z-x}$,and $c = \frac{z}{x-y}$,where $x, y$,and $z$ are distinct such that $x-y, y-z, z-x \neq 0$,then what is the value of $ab + bc + ca + abc$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module