# Solving Trigonometric Equations

Solving trigonometric equations is finding the solutions of equations like we did with linear, quadratic, and radical equations, but using trig functions instead. We will mainly use the Unit Circle to find the exact solutions if we can, and we’ll start out by finding the solutions from $\left[ 0,2\pi \right)$. We can also solve these using a Graphing Calculator, as we’ll see below. Note that we will use Trigonometric Identities to solve trig problems in the Trigonometric Identity section.

Important Note: There is a subtle distinction between finding inverse trig functions and solving for trig functions. If we want $\displaystyle {{\sin }^{{-1}}}\left( {\frac{{\sqrt{2}}}{2}} \right)$ for example, like in the Inverse Trigonometric Functions section, we only pick the answers from Quadrants I and IV, so we get $\displaystyle \frac{\pi }{4}$ only. But if we are solving $\displaystyle \sin \left( x \right)=\frac{{\sqrt{2}}}{2}$ we get $\displaystyle \frac{\pi }{4}$ and $\displaystyle \frac{{3\pi }}{4}$ in the interval ${\left[ {0,2\pi } \right)}$; there are no domain restrictions. In these cases, we want all solutions in the given interval.

## Solving Trigonometric Equations Using the Unit Circle

Let’s start out with solving fairly simple trig equations, and getting the solutions from $\left[ 0,2\pi \right)$, or $\left[ {0,{{{360}}^{o}}} \right)$. Here is the Unit Circle again so we can “pick off” the answers from it:

Notice that we always isolate the trig function, and some solutions may have none, or more than one solution. If there are multiply angles on the unit circle for that trig function, and an expression is involved, we may have to divide up the equation into two separate equations and solve each, like the example with $\displaystyle \theta +\frac{\pi }{{18}}$.

If a square is involved, when we take the square root, we have to include both the positive and negative values. (Note that ${{\left( \cos \theta \right)}^{2}}$  is written as ${{\cos }^{2}}\theta$, and we can put it in the graphing calculator as $\boldsymbol{\cos {{\left( x \right)}^{2}}}$ or $\boldsymbol {{{\left( {\cos \left( x \right)} \right)}^{2}}}$).

Note that sometimes you may have to solve using degrees $\left[ {0,{{{360}}^{o}}} \right)$ instead of radians. The last problem involves solving a trig inequality.

## Solving Trigonometric Equations – General Solutions

Since trig functions go on and on in both directions of the $x$-axis, we’ll also have to know how to solve trig equations over the set of real numbers; this is called finding the general solutions for these equations. We still use the Unit Circle to do this, but we have to think about adding and subtracting multiples of $2\pi$ for the sin, cos, csc, and sec functions (since $2\pi$ is the period for them), and $\pi$ for the tan and cot functions (since $\pi$ is the period for them). We can do this by adding $2\pi k$ or $\pi k$ where $k$ is any integer (positive, negative, or 0); sometimes this can be simplified.

We need to be careful about domain restrictions with our answers. For tan, cot, csc, and sec, we have asymptotes, and if our answer happens to fall on an asymptote, we have to eliminate it.

Here are examples; find the general solution, or all real solutions for the following equations. Note that $k$ represents all integers $\left( k\in \mathbb{Z} \right)$. Note also that I’m using “fancy” notation; you may not be required to do this.

Note that you can check these in a graphing calculator (radian mode) by putting the left-hand side of the equation into ${{Y}_{1}}$ and the right-hand side into ${{Y}_{2}}$ and get the intersection. You won’t get the exact answers, but you can still compare to the exact answers you got above.

## Solving Trigonometric Equations with Multiple Angles

We have to be careful when solving trig equations with multiple angles, meaning there is a coefficient before the $x$ or $\theta$ (variable). This is because we could have fewer or more solutions in the Unit Circle, and thus for all real solutions when we add the $2\pi k$ or $\pi k$. Thus, when we solve these types of trig problems, we always want to solve for the General Solution first (even if we’re asked to get the solutions between 0 and $2\pi k$) and then go back and see how many solutions are on the Unit Circle (between 0 and $2\pi k$ ).

When solving trig equations with multiple angles between 0 and $2\pi$, we’ll typically get fewer solutions if the coefficient of the variable is less than 1, or more solutions if the coefficient of the variable is greater than 1. As an example, we typically get two solutions for $\cos \left( \theta \right)$  between 0 and $2\pi$, so for  $\cos \left( 3\theta \right)$, we’ll get 2 times 3, or 6 solutions. As another example, for $\displaystyle \cos \left( \frac{\theta }{2} \right)$, we’ll only get one solution instead of the normal two. And always check for extraneous solutions. Note that when we multiply or divide to get the variable by itself, we have to do the same with the “$+2\pi k$” or “$+\pi k$”.

Here are some problems: solve the following trig equations for 1) General Solutions, and 2) Solutions between $\left[ {0,2\pi } \right)$ or $\left[ {0,360{}^\circ } \right)$:

## Factoring to Solve Trigonometric Equations

Note that sometimes we have to factor the equations to get the solutions, typically if they are trig quadratic equations. Then we set all factors to 0 to solve, making sure we test the answers to see if they work. We learned how to factor Quadratic Equations in the Solving Quadratics by Factoring and Completing the Square section.

Here are some general hints when solving advanced trig equations:

• When solving, simplify with identities first, if you can.
• You can square each side, but don’t divide both sides by factors with variables, since you might be missing out on solutions. If you need to cross-multiply, or multiply both sides by what’s in a denominator (even when one side equals 0), make sure you’re not missing solutions. (It might be a good idea to see how many solutions there are in a graphing calculator if you can). And always check for extraneous solutions: solutions must work in the original equations, and denominators can’t be 0.
• If you get answers for any trig function that has asymptotes (like tan), check for extraneous solutions (solutions that would be asymptotes).

Here are some examples, both solving on the interval 0 to $2\pi$ (or $360{}^\circ$) and over the reals. In the last problem, the answer ($\displaystyle \theta = \frac{{\pi k}}{2}$) has to be “thrown out”, because of our domain restriction for cot (it falls on an asymptote); this is an extraneous solution:

## Solving Trigonometric Equations Using a Calculator

We already used a calculator to find inverse trig functions here in the Inverse Trigonometric Functions section. When solving trig equations, however, it’s a little more complicated, since typically we’ll have multiple solutions. We can use a scientific calculator, or graph the functions and find intersections with a graphing calculator (usually easier). Don’t forget to change to the appropriate mode (radians or degrees) using DRG on a TI scientific calculator, or mode on a TI graphing calculator.

If just using a scientific calculator, here are some rules for solving trig problems in the intervals $\left[ {0,2\pi } \right)$ or $\left( {-\infty ,\infty } \right)$ in radians (substitute 180° for  if using degrees). Remember these rules, which make sense if you look at the trig functions on the Unit Circle. Remember that $k$ is any integer, negative, 0, or positive.

• For $\displaystyle \sin \theta =A,\,\,\theta ={{\sin }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)$ and also $\displaystyle \theta =\pi -{{\sin }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)$. (For $\csc \theta =A$, use $\displaystyle {{\sin }^{{-1}}}\left( {\frac{1}{A}} \right)$). For example, in the interval $\left[ {0,2\pi } \right)$, for $\displaystyle \sin \theta =.5,\,\,\theta ={{\sin }^{{-1}}}\left( {.5\,} \right)\approx .524\,\,\left( {\frac{{\pi }}{6}\,} \right)$, and also $\displaystyle \theta =\pi -{{\sin }^{{-1}}}\left( {.5\,} \right)\approx 2.618\,\,\left( {\frac{{5\pi }}{6}\,} \right)$.
• For $\displaystyle \cos \theta =A,\,\theta ={{\cos }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)$, and also $\displaystyle \theta =2\pi -{{\cos }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)$, which is the same as $\displaystyle -{{\cos }^{{-1}}}A\,\,\left( {+\,2\pi k} \right)$. (For $\sec \theta =A$, use $\displaystyle {{\cos }^{{-1}}}\left( {\frac{1}{A}} \right)$ ). For example, in the interval $\left[ {0,2\pi } \right)$, for $\displaystyle \cos \theta =.5,\,\,\theta ={{\cos }^{{-1}}}\left( {.5} \right)\,\approx 1.047\,\,\left( {\frac{\pi }{3}} \right)$, and also $\displaystyle \theta =2\pi -{{\cos }^{{-1}}}\left( {.5} \right)\,\approx 5.236\,\,\left( {\frac{{5\pi }}{3}} \right)$.
• For $\displaystyle \tan \theta =A,\,\,\theta ={{\tan }^{{-1}}}A\,\,\left( {+\,\pi k} \right)$; this will find all the solutions. (For $\cot \theta =A$, use $\displaystyle {{\tan }^{{-1}}}\left( {\frac{1}{A}} \right)$, but be careful with the angles $\displaystyle \frac{\pi }{2}+\pi k$, since the calculator shows undefined, instead of 0; you can use $\displaystyle \cot =\frac{{\cos }}{{\sin }}$ instead). For example, in the interval $\left[ {0,2\pi } \right)$, for $\displaystyle \tan \theta =1,\,\,\theta ={{\tan }^{{-1}}}\left( 1 \right)\,\approx .785\,\,\left( {\frac{\pi }{4}} \right)$, and also $\displaystyle \theta ={{\tan }^{{-1}}}\left( 1 \right)+\pi \,\approx 3.927\,\,\left( {\frac{{5\pi }}{4}} \right)$.

Remember that when the coefficient of the argument of the inverse trig functions isn’t 1, we need to divide the $+\,2\pi k$ or $+\,\pi k$ by this coefficient, since the period changes (see examples below). Typically, the default mode is radian mode, unless problem says “degrees”.

If you have access to a graphing calculator, it’s usually easier to solve trig equations. We can put the left-hand part of the equation in ${{Y}_{1}}$, the right-hand part of the equation in ${{Y}_{2}}$, and solve for the intersection(s) between 0 and $2\pi$, or whatever the period when finding general solutions. (Use the trace feature and arrow keys to get close to each intersection, and then use the intersect feature (2nd trace, 5, enter, enter, enter) to find the intersection.) For the reciprocal functions, take the reciprocal of what’s on the right-hand side, and use the regular trig functions.

For intervals of $\left[ {0,2\pi } \right)$, use Xmin = 0, and Xmax = $2\pi$. For general solutions (over the reals), use Xmin = 0 and Xmax = the period (such as $\displaystyle \frac{2\pi }{5}$ when you have $\sin \left( {5x} \right)$, for example) for general solutions. Then, for your answer, add the appropriate factors of $\pi k$, $2\pi k$, or whatever the period of the function is.

Also remember that ${{\left( \cos \theta \right)}^{2}}$ is written as ${{\cos }^{2}}\theta$, and we can put it in the graphing calculator as $\boldsymbol{\cos {{\left( x \right)}^{2}}}$ or $\boldsymbol {{{\left( {\cos \left( x \right)} \right)}^{2}}}$.

Here are some examples using both types of calculators:

## Solving Trigonometric Systems of Equations

Systems of equations are needed when solving for more than one variable in equations. We learned how to solve systems of equations here in the the Systems of Linear Equations and Word Problems section, and systems of more complicated equations here in the Systems of non-Linear Equations section. Again, use either Substitution or Elimination, depending on what’s easier. Once we get the initial solution(s), we’ll can plug in a variable to get the other variable.

Here are some examples of Solving Systems with Trig Equations; solve over the reals:

## Solving Trigonometric Inequalities

Sometimes you might be asked to solve a Trig Inequality. (Links to other types of Inequalities are found here).

We can either solve these inequalities graphically or algebraically; let’s try one of each. Note that you can also solve these on your graphing calculator, using the Intersect feature, and then see where the inequalities “work”:

Practice these problems, and practice, practice, practice!

Hit Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

For Practice: Use the Mathway widget below to try a Trig Solving problem. Click on Submit (the blue arrow to the right of the problem) and click on Solve for x to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to Trigonometric Identities – you’re ready!

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