First slide

Extreme values and Periodicity of Trigonometric functions

Question

Let $a_{1} = \cos (\sin πx), a_{2} = \sin (\cos πx)$ and $a_{3} = \cos (π (x + 1)), where - \frac{1}{2} < x < 0$ .What are the relative sizes of $a_{1}, a_{2} and a_{3}$ ?

Difficult

A

$a_{3} < a_{2} < a_{1}$
B

$a_{1} < a_{3} < a_{2}$
C

$a_{3} < a_{1} < a_{2}$
D

$a_{2} < a_{3} < a_{1}$

Solution

Let y = - x, then we have $0 < y < \frac{1}{2}$
$a_{1} = \cos (\sin πy) > 0, a_{2} = \sin (\cos πy) > 0$ and $a_{3} = \cos (1 - y) π = - \cos πy < 0$
now, $\sin πy + \cos πy \leq \sqrt{2} < \frac{π}{2}$
$\begin{array}{l} \Rightarrow \sin (\cos πy) < \cos (\sin πy) \\ \Rightarrow a_{2} < a_{1} \\ ∴ a_{3} < a_{2} < a_{1} \end{array}$

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