First slide

Trigonometric equations

Question

In which of the following sets the inequality $\sin^{6} x + \cos^{6} x > 5 / 8$ holds good?

Moderate

A

$(- π / 8, π / 8)$
B

$(3 π / 8, 5 π / 8)$
C

$(π / 4, 3 π / 4)$
D

$(7 π / 8, 9 π / 8)$

Solution

${(\sin^{2} x + \cos^{2} x)}^{3} - 3 \sin^{2} {xcos}^{2} x (\sin^{2} x + \cos^{2} x) > \frac{5}{8}$
$\begin{array}{l} or 1 - 3 \sin^{2} {xcos}^{2} x > \frac{5}{8} \\ or \frac{3}{8} > 3 \sin^{2} {xcos}^{2} x \end{array}$
$\begin{array}{l} \Rightarrow 1 - 2 \sin^{2} 2 x > 0 \\ \Rightarrow \cos 4 x > 0 \\ \Rightarrow 4 x \in (- \frac{π}{2}, \frac{π}{2}) \\ \Rightarrow 4 x \in (2 nπ - \frac{π}{2}, 2 nπ + \frac{π}{2}), \end{array}$
$n \in Z$ , generalizing now verify.

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