First slide

Extreme values and Periodicity of Trigonometric functions

Question

$a \leq 5 \cos θ + 3 \cos (θ - \frac{π}{3}) + 8 \leq b$ then

Moderate

A

$a = 1, b = 15$
B

$a = 2, b = 14$
C

$a = 3, b = 13$
D

$a = 4, b = 12$

Solution

$5 \cos θ + 3 \cos (θ - \frac{π}{3}) + 8$
$= 5 \cos θ + 3 (\frac{1}{2} \cos θ + \frac{\sqrt{3}}{2} \sin θ) + 8$
$= \frac{13}{2} \cos θ + \frac{3 \sqrt{3}}{2} \sin θ + 8 m i n i m u m a n d m a x i m u m i s 8 \pm \sqrt{\frac{169 + 27}{4}}$
$∴ a = 8 - 7 = 1, b = 8 + 7 = 15$

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