First slide

Trigonometric Identities

Question

The greatest among $(\sin 1 + \cos 1), (\sqrt{\sin 1} + \sqrt{\cos 1}), (\sin 1 - \cos 1)$ and 1 is

Moderate

A

$\sqrt{\sin 1} + \sqrt{\cos 1}$
B

$\sin 1 + \cos 1$
C

$\sin 1 - \cos 1$
D

1

Solution

$W e h a v e$
$0 < x < 1 \Rightarrow x^{2} < x$
$\begin{array}{l} \sqrt{\sin 1} > \sin 1 > \sin^{2} 1; \sqrt{\cos 1} > \cos 1 > \cos^{2} 1 \\ A d d i n g (\sqrt{\sin 1} + \sqrt{\cos} 1) > (\sin 1 + \cos 1) \\ > (\sin^{2} 1 + \cos^{2} 1) \\ \Rightarrow (\sqrt{\sin 1} + \sqrt{\cos 1}) > (\sin 1 + \cos 1) > 1 \end{array}$

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