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cosx + cosy=3/2 and sinx + siny=1/2 and theta is the A.M. of x and y then sin(2theta)+cos(2theta)=

cosx + cosy=3/2 and sinx + siny=1/2 and theta is the A.M. of x and y then sin(2theta)+cos(2theta)=
 

Grade:12

1 Answers

arun
123 Points
8 years ago
\cos x + \cos y = \frac{3}{2}
2\cos (\frac{x + y}{2})\cos (\frac{x-y}{2}) = \frac{3}{2}             \left [\because \cos x + \cos y = 2\cos (\frac{x + y}{2})\cos (\frac{x-y}{2}) \right ]
and \theta = \frac{x + y}{2}
so, 2\cos (\theta )\cos (\frac{x-y}{2}) = \frac{3}{2}                                       …................................................1
similarly, 2\sin (\theta )\cos (\frac{x-y}{2}) = \frac{1}{2}                              ….................................................2
dividing equation 2 by 1
\tan\theta = \frac{1}{3} 
so, \sin\theta = \frac{1}{\sqrt{10}}
and \cos\theta = \frac{3}{\sqrt{10}}
now using the formula for sin2theta and cos2theta i.e
\sin 2\theta + \cos 2\theta = 2\sin \theta\cos \theta + \cos^{2} \theta - \sin^{2} \theta
\sin 2\theta + \cos 2\theta = \frac{3}{5} + \frac{4}{5}
\sin 2\theta + \cos 2\theta = \frac{7}{5}

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