Question

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

• A. $11,-5$
• B. $3,-3$
• C. 3,0
• D. $10,-4$

Answer 1

Answer: (D)

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