Riemann surface
matlab

Example 1: Find all of the solutions for the equation $2 \cos x=\sqrt{2}$.

Solution:
Isolate the function on one side of the equation
\begin{aligned} & 2 \cos x=\sqrt{2} \ & \cos x=\frac{\sqrt{2}}{2} \end{aligned}
Identify the quadrants for the solutions on the interval $[0,2 \pi)$
Cosine is positive in quadrants I and IV
Solve for the variable
$$x=\frac{\pi}{4}\left(\text { quadrant I) } \quad x=2 \pi-\frac{\pi}{4}=\frac{7 \pi}{4}(\text { quadrant IV) }\right.$$

Example 2: Find all of the solutions for the equation $\tan x=\sqrt{3}$.

Solution:
Identify the quadrants for the solutions on the interval $[0, \pi)$
Note: On this problem we are using the interval $[0, \pi)$ instead of $[0,2 \pi)$ because tangent has a period of $\pi$.
Tangent is positive in quadrants I
Solve for the variable
$$\mathrm{x}=\frac{\pi}{3}$$
Add $n \pi$ to the value of $x$
$$\mathrm{x}=\frac{\pi}{3}+\mathrm{n} \pi$$

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