We will begin by solving trigonometric equations of the form
\begin{align*}
\cos(x) &= a\\
\sin(x) &= a\\
\tan(x) &= a.
\end{align*}
To solve these types of trigonometric equations, we can follow the procedure below.
Solve the equation in quadrant one using the reference angle. You will have to consider $|a|$ if $a<0$.
Use the CAST Rule to determine if there are any other quadrants you need to consider. Solve the equation in these quadrants as well.
If you have a specific interval for $x$, ensure all values of $x$ fall within this interval.
If you do not have a specific interval for $x$, you must use a general solution. For $\sin(x)$ and $\cos(x)$, we add ``$+2k\pi$'' to each solution. For $\tan(x)$ we add ``$+k\pi$'' to each solution.
Basic trigonometric equations, part 1.
Basic trigonometric equations, part 2.
Reciprocal Trigonometric Equations
We will next see how to solve trigonometric equations of the form
$$\begin{align*}
\sec(x) &= a\\
\csc(x) &= a\\
\cot(x) &= a.
\end{align*}$$
To solve these types of trigonometric equations, we can follow the procedure below.
Change the equation into a basic trigonometric equation using one of the reciprocal identities:
\begin{align*}
\sec(x) &= \frac{1}{\cos(x)}\\
\csc(x) &= \frac{1}{\sin(x)}\\
\cot(x) &= \frac{1}{\tan(x)}.
\end{align*}
Solve the new equation using the method described above.
Reciprocal trigonometric equations, part 1.
Equations Involving Multiple Angles
A multiple angle equation is one involving a multiple of the variable within the trigonometric function (like $2x$ or $3x$). To solve these types of trigonometric equations, we can follow the procedure below.
If the angle of your trigonometric function is $bx$, make a substitution $\alpha = bx$.
Now you should have either a basic trigonometric equation, or a reciprocal trigonometric equation.
Solve this equation for $\alpha$ using the methods outlined above. \textbf{Ensure that you form the most general solution.}
Replace $\alpha$ with $bx$ and divide all solutions by $b$.
If you do not have a specific interval for $x$, you now have your most general solution and you are done.
If you do have an interval for $x$, substitute values of $k$ into your general solution to determine the specific values of $x$ that lie in the given interval.
Multiple angle equations, part 1.
Multiple angle equations, part 2.
Other Techniques
The videos below outline some other common techniques to be aware of when solving trigonometric equations.
Factoring.
Using an identity.