Solve the trigonometric equation for all values $$ 0 \leq x

Question

Solve the trigonometric equation for all values $$ 0 \leq x<2 \pi $$ $$ 2 \cos x=-\sqrt{2} $$

UpStudy AI · Accepted Answer

1. Divide both sides of the equation by 2:
   $$
   2 \cos x = -\sqrt{2} \quad \Rightarrow \quad \cos x = -\frac{\sqrt{2}}{2}.
   $$

2. Recall that the cosine function equals $$-\frac{\sqrt{2}}{2}$$ at the angles:
   $$
   x = \frac{3\pi}{4} \quad \text{and} \quad x = \frac{5\pi}{4},
   $$
   since these are the angles in the interval $$0 \leq x < 2\pi$$ where the cosine value is negative.

3. Therefore, the solutions to the equation in the interval $$0 \leq x < 2\pi$$ are:
   $$
   x = \frac{3\pi}{4} \quad \text{or} \quad x = \frac{5\pi}{4}.
   $$
The solutions are $$ x = \frac{3\pi}{4} $$ and $$ x = \frac{5\pi}{4} $$.

Solve the trigonometric equation for all values

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