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  4. The greatest and least value of y=3 cos (θ+(π/3))+5 cos θ+3 are respectively

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Q. The greatest and least value of $y=3 \cos \left(\theta+\frac{\pi}{3}\right)+5 \cos \theta+3$ are respectively

Trigonometric Functions

Solution:

$y=3 \cos \left(\theta+\frac{\pi}{3}\right)+5 \cos \theta+3$
$y=3 \cos \theta \cdot \frac{1}{2}-3 \frac{\sqrt{3}}{2} \sin \theta+5 \cos \theta+3$
$y=\frac{3}{2} \cos \theta-\frac{3 \sqrt{3}}{2} \sin \theta+5 \cos \theta+3$
$y=\frac{13}{2} \cos \theta-\frac{3 \sqrt{3}}{2} \sin \theta+3$
$y_{\max }=\sqrt{\frac{169}{4}+\frac{27}{4}}+3=7+3=10$
$y_{\min }=-\sqrt{\frac{169}{4}+\frac{27}{4}}+3=-7+3=-4$