First slide

Multiple and sub- multiple Angles

Question

Let $α and, β$ be such that $π < α - β < 3 π . If \sin α + \sin β = \frac{21}{65}$ and $\cos α + \cos β = - \frac{27}{65},$ then the value of $\cos \frac{α - β}{2}$ is

Moderate

A

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

$\frac{3}{\sqrt{130}}$
C

$\frac{6}{65}$
D

$- \frac{6}{65}$

Solution

We have $\sin α + \sin β = \frac{21}{65} - - - - - (i)$
$\cos α + \cos β = - \frac{27}{65} - - - - (ii)$
Squaring Eq. (i), we get
$\sin^{2} α + \sin^{2} β + 2 \sin α \sin β = {(\frac{21}{65})}^{2} - - - (i i i)$
Squaring Eq. (ii), we get
$\cos^{2} α + \cos^{2} β + 2 \cos αcos β = {(\frac{27}{65})}^{2} - - - - (iv)$
Adding Eqs. (iii) and (iv), we get
$\begin{aligned} 2 + 2 \cos (α - β) = \frac{1}{(65)^{2}} [(27)^{2} + (21)^{2}] \\ = \frac{1}{(65)^{2}} (729 + 441) \\ = \frac{1}{(65)^{2}} (1170) = \frac{18}{65} \end{aligned}$
$\begin{array}{l} or 1 + \cos (α - β) = \frac{9}{65} \\ or 2 \cos^{2} \frac{α - β}{2} = \frac{9}{65} \\ or \cos \frac{α - β}{2} = - \frac{3}{\sqrt{130}} \end{array}$
$[∵ π < α - β < 3 π \Rightarrow \frac{π}{2} < \frac{α - β}{2} < \frac{3 π}{2} \Rightarrow \cos (\frac{α - β}{2}) < 0]$

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