Question

sin^2x+2cosx=2

Answer 1

Task:

$$\sin^ {2} x + 2 \cos x = 2$$

Solution:

$$\sin^ {2} x + 2 \cos x = 2$$

From the basic relationship between the sine and the cosine $\sin^2 x + \cos^2 x = 1$ , we get: $\sin^2 x = 1 - \cos^2 x$ .

$$1 - \cos^ {2} x + 2 \cos x = 2$$ $$0 = 2 - (1 - \cos^ {2} x + 2 \cos x)$$ $$2 - 1 + \cos^ {2} x - 2 \cos x = 0$$ $$\cos^ {2} x - 2 \cos x + 1 = 0$$

We make the substitution $t = \cos x$ :

$$t ^ {2} - 2 t + 1 = 0$$ $$(t - 1) ^ {2} = 0$$ $$t = 1$$

So $\cos x = t = 1$ . Let's find $x$ inverse cosine:

$$x = \cos^ {- 1} 1$$ $$x = 2 \pi n, n \in \mathbb {Z}$$

$\mathbb{Z}$ is the set of integer numbers.

Answer: $x = 2\pi n$