For all values of $ heta$, the values of $3-\cos heta+\cos \left( heta+\frac{\pi}{3} ight)$ lie in the interval :

Question

For all values of $	heta$, the values of $3-\cos 	heta+\cos \left(	heta+\frac{\pi}{3}ight)$ lie in the interval :

Aicharya · Accepted Answer

$3-\cos 	heta+\cos \left(	heta+\frac{\pi}{3}ight)$ $=3-\cos 	heta+\cos 	heta \cdot \cos \frac{\pi}{3}-\sin 	heta \cdot \sin \frac{\pi}{3}$ $=3-\cos 	heta+\frac{1}{2} \cos 	heta-\frac{\sqrt{3}}{2} \sin 	heta$ $=3-\frac{1}{2} \cos 	heta-\frac{\sqrt{3}}{2} \sin 	heta$ $=3-\left[\sin \frac{\pi}{6} \cos 	heta+\cos \frac{\pi}{6} \sin 	hetaight]$ $=3-\sin \left(	heta+\frac{\pi}{6}ight)$ Since $-1 \leq \sin 	heta \leq 1$ $	herefore$ The value of expression lies in $[2,4]$.

For all values of $\theta$, the values of $3-\cos \theta+\cos \left(\theta+\frac{\pi}{3}\right)$ lie in the…

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