# Solving Trigonometric Equations

### INTRO

At this point, you should already be familiar with finding the sine or cosine of an angle. Now, let's turn our attention to finding angles that satisfy a trigonometric equation. Let's look at an example of this graphically.

The red curve below represents the function $$y = \sin{(x)}$$ and the blue curve represents the line $$y=1/2$$. Every point at which the blue line intersects the red curve represents a solution of $$\sin{(x)} =1/2$$. Notice how in just this short interval depicted in the graph there are 8 different solutions for $$x$$.

Depending on whether the right-hand side of the equation is a positive or negative number will determine which quadrants the angles will be in. Since the question above is dealing with sine and the right-hand side is positive, we know that all the solutions for $$x$$ will be angles in the first and second quadrant. Then using the special triangles and unit circle, we can determine the possible values of $$x$$.

Example: Solve $$\sin(x) = \frac{1}{2}$$ for $$x$$ on the interval $$[-2\pi,2\pi]$$.

Solution: $$x = \frac{\pi}{6}, \frac{5\pi}{6}$$. Note, that we are using units of radians (since the interval is also in radians).

Also, we can find these angles going clockwise:

$$x = \frac{-7\pi}{6}, \frac{-11\pi}{6}$$.

Example: Solve $$\cos(\frac{x}{2}) = 1$$ for $$x$$.

Solution: $$\cos(\frac{x}{2}) = 1$$.

So $$(\frac{x}{2}) = 2n\pi$$ where we have chosen to solve for $$x$$ in units of radians.

Consequently, $$x = 4n\pi$$, where $$n$$ is any integer.

Example 1:

Example 2:

Example 3: