First slide

Trigonometric equations

Question

The number of solutions of the equation $\sin^{3} x \cos x + \sin^{2} x \cos^{2} x + \sin x \cos^{3} x = 1$ in the interval [0,2 $π$ ] is/ are

Moderate

A

0
B

2
C

3
D

infinite

Solution

$\sin^{3} xcos x + \sin^{2} {xcos}^{2} x + \sin {xcos}^{3} x = 1$
or $\sin xcos x (\sin^{2} x + \sin xcos x + \cos^{2} x) = 1$
or $\frac{\sin 2 x}{2} (1 + \frac{\sin 2 x}{2}) = 1$
or $\sin 2 x (2 + \sin 2 x) = 4$
or $\sin^{2} 2 x + 2 \sin 2 x - 4 = 0$
or $\sin 2 x = \frac{- 2 \pm \sqrt{4 + 16}}{2} = - 1 \pm \sqrt{5}$
This is not possible since $- 1 \leq \sin 2 x \leq 1$
Hence, the given equation has no solution.

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