Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa

Let \alpha ,\beta   be such that  \pi < \alpha -\beta < 3\pi .If  \sin \alpha +\sin \beta = -21/65,  and 

\cos \alpha +\cos \beta = -27/65 then the value of  \cos \frac{\alpha -\beta }{2} is :

  • Option 1)

    \frac{6}{65}

  • Option 2)

    \frac{3}{\sqrt{130}}

  • Option 3)

    -\frac{3}{\sqrt{130}}

  • Option 4)

    \frac{-6}{65}

 

Answers (1)

best_answer

As we learnt in 

Trigonometric Ratios of Submultiples of an Angle -

Trigonometric ratios of submultiples of an angle 1

- wherein

This shows the formulae for half angles and their doubles.

 

 \sin \alpha +\sin \beta =\frac{-21}{65}\Rightarrow 2\sin \frac{(\alpha +\beta )}{2}\cos \frac{(\alpha -\beta )}{2}=\frac{-21}{65}    -----------------(i)

\cos \alpha +\cos \beta =\frac{-27}{65}\Rightarrow 2\cos \frac{(\alpha +\beta )}{2}\cos \frac{(\alpha -\beta )}{2}=\frac{-21}{65} -------------------(ii)

Squaring and adding,

4\cos ^{2}\frac{(\alpha -\beta) }{2}[\sin ^{2}\frac{\alpha +\beta}{2} +\cos ^{2}\frac{\alpha +\beta}{2}]=\frac{21^{2}+27^{^{2}}}{65^{2}}

\Rightarrow \cos ^{2}\frac{(\alpha -\beta) }{2}=\frac{1170}{4\times 4225}\Rightarrow \cos ^{2}\frac{(\alpha -\beta) }{2}=\frac{1170}{4\times 4225}=\frac{9}{130}

\Rightarrow \cos\frac{(\alpha -\beta) }{2}=\frac{-3}{\sqrt{130}}

[\because \pi< \alpha -\beta < 3\pi\Rightarrow \frac{\pi}{2} \leq \frac{\alpha-\beta}{2} \leq \frac{3\pi}{2} \Rightarrow cos\left(\frac{\alpha-\beta}{2} \right )<0]

 


Option 1)

\frac{6}{65}

This is incorrect option

Option 2)

\frac{3}{\sqrt{130}}

This is incorrect option

Option 3)

-\frac{3}{\sqrt{130}}

This is correct option

Option 4)

\frac{-6}{65}

This is incorrect option

Posted by

divya.saini

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2024 Topper: Farmer’s son Gajare Nilkrishna says ‘keep questioning until you understand’
anam-khan

NIT Puducherry Cut-off 2024: Computer Science and Engineering had highest closing rank last year
anam-khan

JEE Mains: Farmer's son from remote Maharashtra village aces exam

Latest Question