First slide

Extreme values and Periodicity of Trigonometric functions

Question

Given $A = \sin^{2} θ + \cos^{4} θ$ , then for all real $θ$

Moderate

A

$1 \leq A \leq 2$
B

$\frac{3}{4} \leq A \leq 1$
C

$\frac{13}{16} \leq A \leq 1$
D

$\frac{3}{4} \leq A \leq \frac{13}{16}$

Solution

We have,
$\begin{array}{l} A = \sin^{2} θ + \cos^{4} θ \\ \Rightarrow A = \frac{1 - \cos 2 θ}{2} + {(\frac{1 + \cos 2 θ}{2})}^{2} \\ \Rightarrow A = \frac{1}{2} - \frac{1}{2} \cos 2 θ + \frac{1}{4} + \frac{1}{2} \cos 2 θ + \frac{1}{4} \cos^{2} 2 θ \end{array}$
$\begin{array}{l} \Rightarrow A = \frac{3}{4} + \frac{1}{4} \{\frac{\cos 4 θ + 1}{2}\} = \frac{3}{4} + \frac{1}{8} + \frac{1}{8} \cos 4 θ \\ Now, - 1 \leq \cos 4 θ \leq 1 \\ \Rightarrow \frac{- 1}{8} \leq \frac{\cos 4 θ}{8} \leq \frac{1}{8} \\ \Rightarrow \frac{3}{4} + \frac{1}{8} - \frac{1}{8} \leq \frac{3}{4} + \frac{1}{4} \{\frac{1 + \cos 4 θ}{2}\} \leq \frac{3}{4} + \frac{1}{8} + \frac{1}{8} \\ \Rightarrow \frac{3}{4} \leq A \leq 1 \end{array}$

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