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)
D Divya Saini

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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