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Trigonometry> cosx + cosy=3/2 and sinx + siny=1/2 and t...

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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)=

mahi , 9 Years ago

Grade 12

1 Answers

arun

$\cos x + \cos y = \frac{3}{2}$

$2\cos (\frac{x + y}{2})\cos (\frac{x-y}{2}) = \frac{3}{2}$ $\left [\because \cos x + \cos y = 2\cos (\frac{x + y}{2})\cos (\frac{x-y}{2}) \right ]$

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

$\sin 2\theta + \cos 2\theta = 2\sin \theta\cos \theta + \cos^{2} \theta - \sin^{2} \theta$

$\sin 2\theta + \cos 2\theta = \frac{3}{5} + \frac{4}{5}$

$\sin 2\theta + \cos 2\theta = \frac{7}{5}$

Last Activity: 9 Years ago

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