First slide

Trigonometric equations

Question

If $0 \leq x \leq 2 π$ then the number of solutions of $3 (\sin x + \cos x) - 2 (\sin^{3} x + \cos^{3} x) = 8$ is

Moderate

A

0
B

1
C

2
D

4

Solution

The given equation is
$\begin{array}{ll} 3 (\sin x + \cos x) - 2 (\sin x + \cos x) (1 - \sin xcos x) = 8 \\ or (\sin x + \cos x) [3 - 2 + 2 \sin xcos x] = 8 \\ or (\sin x + \cos x) [\sin^{2} x + \cos^{2} x + 2 \sin xcos x] = 8 \\ or (\sin x + \cos x)^{3} = 8 \\ or \sin x + \cos x = 2 \end{array}$
or sinx, cosx= 1, which is not possible. Hence, the given equation has no solution.

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