The minimum value of cos 2theta + cos theta f | vedclass

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Question

(A) Let $S = \cos 2	heta + \cos 	heta$.Using the identity $\cos 2	heta = 2\cos^2 	heta - 1$,we get:$S = 2\cos^2 	heta - 1 + \cos 	heta$.To find

Vedclass · Accepted Answer

$-9/8$

Vedclass · Answer

(A) Let $S = \cos 2	heta + \cos 	heta$.Using the identity $\cos 2	heta = 2\cos^2 	heta - 1$,we get:$S = 2\cos^2 	heta - 1 + \cos 	heta$.To find the minimum value,we complete the square for the quadratic expression in $\cos 	heta$:$S = 2(\cos^2 	heta + \frac{1}{2}\cos 	heta) - 1$.$S = 2(\cos^2 	heta + \frac{1}{2}\cos 	heta + \frac{1}{16} - \frac{1}{16}) - 1$.$S = 2(\cos 	heta + \frac{1}{4})^2 - \frac{1}{8} - 1$.$S = 2(\cos 	heta + \frac{1}{4})^2 - \frac{9}{8}$.Since the square term $2(\cos 	heta + \frac{1}{4})^2 \geq 0$,the minimum value of $S$ is attained when $\cos 	heta = -1/4$.Thus,the minimum value is $-9/8$.

The minimum value of $\cos 2\theta + \cos \theta$ for real values of $\theta$ is

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