# Solving Trigonometric Equations

## Key Questions

• Solving concept. To solve a trig equation, transform it into one, or many, basic trig equations. Solving a trig equation, finally, results in solving various basic trig equations.
There are 4 main basic trig equations:
sin x = a; cos x = a; tan x = a; cot x = a.
Exp. Solve sin 2x - 2sin x = 0
Solution. Transform the equation into 2 basic trig equations:
2sin x.cos x - 2sin x = 0
2sin x(cos x - 1) = 0.
Next, solve the 2 basic equations: sin x = 0, and cos x = 1.
Transformation process.
There are 2 main approaches to solve a trig function F(x).
1. Transform F(x) into a product of many basic trig functions.
Exp. Solve F(x) = cos x + cos 2x + cos 3x = 0.
Solution. Use trig identity to transform (cos x + cos 3x):
F(x) = 2cos 2x.cos x + cos 2x = cos 2x(2cos x + 1 ) = 0.
Next, solve the 2 basic trig equations.
2. Transform a trig equation F(x) that has many trig functions as variable, into a equation that has only one variable. The common variables to be chosen are: cos x, sin x, tan x, and tan (x/2)
Exp Solve ${\sin}^{2} x + {\sin}^{4} x = {\cos}^{2} x$
Solution. Call cos x = t, we get
$\left(1 - {t}^{2}\right) \left(1 + 1 - {t}^{2}\right) = {t}^{2}$.
Next, solve this equation for t.
Note . There are complicated trig equations that require special transformations.

• As a general description, there are 3 steps. These steps may be very challenging, or even impossible, depending on the equation.

Step 1: Find the trigonometric values need to be to solve the equation.
Step 2: Find all 'angles' that give us these values from step 1.
Step 3: Find the values of the unknown that will result in angles that we got in step 2.

(Long) Example
Solve: $2 \sin \left(4 x - \frac{\pi}{3}\right) = 1$

Step 1: The only trig function in this equation is $\sin$.
Sometimes it is helpful to make things look simpler by replacing, like this:
Replace $\sin \left(4 x - \frac{\pi}{3}\right)$ by the single letter $S$. Now we need to find $S$ to make $2 S = 1$. Simple! Make $S = \frac{1}{2}$
So a solution will need to make $\sin \left(4 x - \frac{\pi}{3}\right) = \frac{1}{2}$

Step 2: The 'angle' in this equation is $\left(4 x - \frac{\pi}{3}\right)$. For the moment, let's call that $\theta$. We need $\sin \theta = \frac{1}{2}$
There are infinitely many such $\theta$, we need to find them all.

Every $\theta$ that makes $\sin \theta = \frac{1}{2}$ is coterminal with either $\frac{\pi}{6}$ or with $\frac{5 \pi}{6}$. (Go through one period of the graph, or once around the unit circle.)
So $\theta$ Which, remember is our short way of writing $4 x - \frac{\pi}{3}$ must be of the form: $\theta = \frac{\pi}{6} + 2 \pi k$ for some integer $k$ or of the form $\theta = \frac{5 \pi}{6} + 2 \pi k$ for some integer $k$.

Step 3:
Replacing $\theta$ in the last bit of step 2, we see that we need one of: $4 x - \frac{\pi}{3} = \frac{\pi}{6} + 2 \pi k$ for integer $k$
or $4 x - \frac{\pi}{3} = \frac{5 \pi}{6} + 2 \pi k$ for integer $k$.

Adding $\frac{\pi}{3}$ in the form $\frac{2 \pi}{6}$ to both sides of these equations gives us:
$4 x = \frac{3 \pi}{6} + 2 \pi k = \frac{\pi}{2} + 2 \pi k$ for integer $k$ or
$4 x = \frac{7 \pi}{6} + 2 \pi k$ for integer $k$.

Dividing by $4$ (multiplying by $\frac{1}{4}$) gets us to:

$x = \frac{\pi}{8} + \frac{2 \pi k}{4}$ or
$x = \frac{7 \pi}{24} + \frac{2 \pi k}{4}$ for integer $k$.

We can write this in simpler form:
$x = \frac{\pi}{8} + \frac{\pi}{2} k$ or
$x = \frac{7 \pi}{24} + \frac{\pi}{2} k$ for integer $k$.

Final note The Integer $k$ could be a positive or negative whole number or 0. If $k$ is negative, we're actually subtracting from the basic solution.

• Examples of trig expressions: # f(x) = sin 2x + cos x; #
$f \left(x\right) = \sin x + \sin 2 x + \sin 3 x$
Examples of trig equations: $f \left(x\right) = \sin 2 x + \cos x = 0$
$f \left(x\right) = \sin x + \sin 2 x + \sin 3 x = 0$
Examples of trig inequalities $f \left(x\right) = \sin 2 x + \cos x > 0$
$f \left(x\right) = \sin x + \sin 2 x + \sin 3 x < 0$

Use trig Transformation Identities to transform these above trig expressions into trig basic expressions, or expressions in simplest form.
Example: Transform $f \left(x\right) = \sin 2 x + \cos x$. Use Identity $\left(\sin 2 a = 2 \sin a \cdot \cos a\right)$ to transform $f \left(x\right) .$
$f \left(x\right) = 2 \cdot \sin x \cdot \cos x + \cos x = \cos x \cdot \left(2 \sin x + 1\right)$
This is f(x) expressed in simplest form.
Trig equation in simplest form: $f \left(x\right) = \cos x \cdot \left(\sin 2 x + 1\right) = 0$
Trig inequality in simplest form: $f \left(x\right) = \cos x \cdot \left(2 \sin x + 1\right) > 0$