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If $\cos \alpha \: +\: \cos \beta \: =\frac{3}{2} \: and\: \sin \alpha +\sin \beta =\frac{1}{2}\; and\; \theta$ is the arithmetic mean of $\alpha$ and $\beta$ , then

$\sin2\theta +\cos 2\theta$ is equal to :

Option 1)
$\frac{3}{5}$

Option 2)
$\frac{4}{5}$

Option 3)
$\frac{7}{5}$

Option 4)
$\frac{8}{5}$

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.

$cos\alpha +cos\beta = \frac{3}{2}$ $\Rightarrow 2 cos\frac{\alpha +\beta }{2} cos \frac{\alpha -\beta }{2 } = \frac{3}{2}\ \: \: \: \: \: \: \: \: \: ................(1)$

$sin\alpha +sin\beta = \frac{1}{2}$ $\Rightarrow 2 sin\frac{\alpha +\beta }{2}\:\:cos\frac{\alpha -\beta }{2}=\frac{1}{2}\ \, \, \, \, \, \, \, \, .........(2)$

dividing (2) by (1), we get

$tan\frac{\left ( \alpha +\beta \right )}{2}=\frac{1}{3}$

$\Rightarrow tan\left ( \alpha +\beta \right )=\frac{2\times \frac{1}{3}}{1-\frac{1}{9}}=\frac{3}{4}$

$\Rightarrow sin \left ( \alpha +\beta \right )=\frac{3}{5},\:\:\:cos\left ( \alpha +\beta \right ) = \frac{4}{5}$

Thus, $sin \left ( \alpha +\beta \right )+cos\left ( \alpha +\beta \right )=\frac{7}{5}$

$\Rightarrow sin2\theta +cos2\theta =\frac{7}{5}$

Option 1)

$\frac{3}{5}$

This option is incorrect.

Option 2)

$\frac{4}{5}$

This option is incorrect.

Option 3)

$\frac{7}{5}$

This option is correct.

Option 4)

$\frac{8}{5}$

This option is incorrect.

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divya.saini

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