Question

Find all solutions to the equation.

7 sin^2x - 14 sin x + 2 = -5

How can I solve this please

Answer 1

Answer on Question #42394 - Math - Geometry

Find all solutions to the equation

$$7 \sin^ {2} x - 14 \sin x + 2 = -5.$$

Solution. First, let us bring all the terms to the right-hand side of the equation:

$$7 \sin^ {2} x - 14 \sin x + 7 = 0$$

and divide by 7:

$$\sin^ {2} x - 2 \sin x + 1 = 0.$$

This can be written as

$$(\sin x - 1) ^ {2} = 0,$$

or

$$\sin x - 1 = 0,$$ $$\sin x = 1.$$

Thus,

$$x = \arcsin 1 + 2\pi n, \quad n \in \mathbb{Z},$$

and finally

$$x = \frac {\pi}{2} + 2 \pi n, \qquad n \in \mathbb {Z}.$$

Answer. $x = \frac{\pi}{2} + 2\pi n$ for any $n \in \mathbb{Z}$.

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