First slide

Trigonometric Identities

Question

Let $θ \in (π / 4, π / 2)$ , which of the following statements is true?

Moderate

A

$(\cos θ)^{\cos θ} < (\sin θ)^{\cos θ} < (\cos θ)^{\sin θ}$
B

$(\cos θ)^{\sin θ} < (\cos θ)^{\cos θ} < (\sin θ)^{\cos θ}$
C

$(\sin θ)^{\cos θ} < (\cos θ)^{\cos θ} < (\cos θ)^{\sin θ}$
D

$(\cos θ)^{\cos θ} < (\cos θ)^{\sin θ} < (\sin θ)^{\cos θ}$

Solution

For $θ \in (π / 4, π / 2), 0 < \cos θ < 1 / \sqrt{2} < \sin θ < 1$
$\Rightarrow (\cos θ)^{\cos θ} < (\sin θ)^{\cos θ}$ and $(\cos θ)^{\cos θ} > (\cos θ)^{\sin θ}$
(as $0 < \cos θ < 1$ and $\cos θ < \sin θ$ )
Showing that (b) is correct.

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