First slide

Trigonometric equations

Question

The values of x between 0 and $2 π$ which satisfy the equation $\sin x \sqrt{8 \cos^{2} x} = 1$ are in A.P. The common difference of the A.P. is

Moderate

A

$π / 8$
B

$π / 4$
C

$3 π / 8$
D

$5 π / 8$

Solution

From the given equation we have $2 \sin x | \cos x | = 1 / \sqrt{2} \Rightarrow \sin 2 x = 1 / \sqrt{2}$ if $\cos x > 0$ and $\sin 2 x = - 1 / \sqrt{2}$ if $\cos x < 0$
$∴$ when $\cos x > 0, \sin 2 x = 1 / \sqrt{2} \Rightarrow x = π / 8, 3 π / 8$
when $\cos x < 0, \sin 2 x = - 1 / \sqrt{2} \Rightarrow x = 5 π / 8, 7 π / 8$ . so the required values of x are
$π / 8, 3 π / 8, 5 π / 8, 7 π / 8$
which form an A.P. with common difference $π / 4$ .

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