Question

Hey !!

Question :-

Choose the correct option from the options given below !
If cos ( α/3 ) = 1/2 find the value of sin α .

OPTION A : 1
OPTION B : 0
OPTION C : -1
OPTION D : 0.5

Also give me any formula that you used in the above problem :) It will be helpful !

DON'T DARE TO SPAM !

Answer 1

Answer:

$\boxed{\boxed{\green{\bold{OPTION\;B}}}}$

Given

$\mathsf{cos(\dfrac{\alpha}{3})=\dfrac{1}{2}}$

Here is a trigonometric identity that will be effective :

$\boxed{\mathtt{cos3A=4cos^3A-3cosA}}$

$\mathsf{cos3x=4cos^3x-3cosx}\\\\\bf{Taking\:\dfrac{\alpha}{3}=x\:in\:the\:equation\:we\:get:}\\\\\implies \mathsf{cos\alpha=4cos^3(\dfrac{\alpha}{3})-3cos\dfrac{\alpha}{3}}\\\\\implies \mathsf{cos\alpha=4(\dfrac{1}{2})^3-3\times \dfrac{1}{2}}\\\\\implies \mathsf{cos\alpha=4\times \dfrac{1}{8}-\dfrac{3}{2}}\\\\\implies \mathsf{cos\alpha=\dfrac{1}{2}-\dfrac{3}{2}}\\\\\implies \mathsf{cos\alpha=\dfrac{1-3}{2}}\\\\\implies \mathsf{cos\alpha=\dfrac{-2}{2}}\\\\\implies \mathsf{cos\alpha=-1}$

Now this is pretty simple :

$\bf{Using\:sin^2A+cos^2A=1\:we\:will\:get\::}\\\\\mathsf{sin^2\alpha+(-1)^2=1}\\\\\implies \mathsf{sin^2\alpha+1=1}\\\\\implies \mathsf{sin^2\alpha=0}\\\\\implies \mathsf{sin\alpha=0}$

OPTION B is the correct one .

NOTE :

Problem is solved . But remember this :

ALL-SIN-TAN-COS Rule .

All ratios are positive in the first quadrant .

The ratio of sine is positive in the second quadrant .

The ratio of tan is positive in the third quadrant .

The ratio of cos is positive in the fourth quadrant .

Answer 2

$\underline{ \sf{ Solution}} :$

$\green{\underline{ \mathcal{Here,}} }$

$\sf{ Cos \: \frac{ \alpha }{3} = \frac{1}{2} }$

$\mathcal \green{\underline{Then,}}$

$\sf Sin \frac{ \alpha }{3} = \sqrt{1 - {Cos}^{2} \frac{ \alpha }{3} } = \sqrt{1 - ( { \frac{1}{2} })^{2} }$

$\sf \implies \sqrt{ \frac{4 - 1}{4} } = \frac{ \sqrt{3} }{2}$

$\boxed {\pink{ \texttt{Now,}}}$

$\sf Sin \: \alpha = 3 \: Sin \frac{ \alpha }{3} - 4 \: {Sin}^{3} \frac{ \alpha }{3}$

$\sf{ \implies 3 \: . \: \frac{ \sqrt{3} }{2} } - 4 (\frac{ \sqrt{3} }{2} ) ^{3}$

$\sf \implies \frac{3 \sqrt{3} }{2} - 4 \: . \: \frac{3 \sqrt{3} }{8}$

$\sf \implies \frac{3 \sqrt{3} }{2} - \frac{3 \sqrt{3} }{2}$

$\sf \implies \: 0$

$\purple {\boxed { \green{ \mathcal{Hence, The \: value \: of \: Sin \: \alpha \: i s \: 0.}}}}$