First slide

Extreme values and Periodicity of Trigonometric functions

Question

If the equation $k \cos x - 3 \sin x = k + 1$ has a solution for $x$ then

Moderate

A

$0 \leq k \leq 8$
B

$k \geq 4$
C

$2 k \leq 1$
D

$k \leq 4$

Solution

$I f k \cos x - 3 \sin x = k + 1 h a s a s o l u t i o n, t h e n - \sqrt{k^{2} + 9} \leq k + 1 \leq \sqrt{k^{2} + 9} \Rightarrow |k + 1| \leq \sqrt{k^{2} + 9} \Rightarrow {(k + 1)}^{2} \leq k^{2} + 9 \Rightarrow k \leq 4$

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