A Negative Cosine

Solve $2\cos\left(x\right)+\sqrt{2} = 0$ on [0°, 360°). Move the √2 across and divide by 2: $\cos\left(x\right) = \frac{-\sqrt{2}}{2}$. The size √2/2 gives the reference angle 45°, and the sign says where: cosine is negative on the left half, quadrants II and III.

Place 45° in each: $180^{\circ}-45^{\circ}$ is 135° and $180^{\circ}+45^{\circ}$ is 225°. Both points sit at x = −√2/2, mirror images across the x axis.