First slide

Extreme values and Periodicity of Trigonometric functions

Question

$Let A = \sin^{8} θ + \cos^{14} θ,then for all real θ$

Easy

A

$A \geq 1$
B

$0 < A \leq 1$
C

$\frac{1}{2} < A \leq \frac{3}{2}$
D

$0 < A \leq \frac{\sqrt{3}}{2}$

Solution

$We know that 0 \leq \sin^{2} θ \leq 1$
$\begin{array}{l} \Rightarrow \sin^{8} θ \leq \sin^{2} θ \\ A l s o \cos^{14} θ \leq \cos^{2} θ \end{array}$
On adding
$\cos^{14} θ + \sin^{8} θ \leq 1$
$∴ 0 < A \leq 1$

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