Solving Trigonometric Equations

Thales of Miletus (circa 625–547 BC) is known as the founder of geometry. The legend is that he calculated the height of the Great Pyramid of Giza in Egypt using the theory of similar triangles, which he developed by measuring the shadow of his staff. Based on proportions, this theory has applications in a number of areas, including fractal geometry, engineering, and architecture. Often, the angle of elevation and the angle of depression are found using similar triangles.

In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as finding the dimensions of the pyramids.

Solving Linear Trigonometric Equations in Sine and Cosine

Trigonometric equations are, as the name implies, equations that involve trigonometric functions. Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all. Often we will solve a trigonometric equation over a specified interval. However, just as often, we will be asked to find all possible solutions, and as trigonometric functions are periodic, solutions are repeated within each period. In other words, trigonometric equations may have an infinite number of solutions. Additionally, like rational equations, the domain of the function must be considered before we assume that any solution is valid. The period of both the sine function and the cosine function is $2\pi$. In other words, every $2\pi$ units, the $y$-values repeat. If we need to find all possible solutions, then we must add $2\pi k$, where $k$ is an integer, to the initial solution. Recall the rule that gives the format for stating all possible solutions for a function where the period is $2\pi$:

There are similar rules for indicating all possible solutions for the other trigonometric functions. Solving trigonometric equations requires the same techniques as solving algebraic equations. We read the equation from left to right, horizontally, like a sentence. We look for known patterns, factor, find common denominators, and substitute certain expressions with a variable to make solving a more straightforward process. However, with trigonometric equations, we also have the advantage of using the identities we developed in the previous sections.

Example. Find all possible exact solutions for the equation $\cos\theta=\tfrac12$.

Solution. From the unit circle, we know that

These are the solutions in the interval $[0,2\pi]$. All possible solutions are given by

where $k$ is an integer.

Example. Find all possible exact solutions for the equation $\sin t=\tfrac12$.

Solution. Solving for all possible values of $t$ means that solutions include angles beyond the period of $2\pi$. From the unit circle, we can see that the solutions are $\tfrac{\pi}{6}$ and $\tfrac{5\pi}{6}$. But the problem is asking for all possible values that solve the equation. Therefore, the answer is

where $k$ is an integer.

Example. Solve the equation exactly: $2\cos\theta-3=-5,\ 0\le\theta<2\pi$.

Solution. Use algebraic techniques to solve the equation.

Solving Equations Involving a Single Trigonometric Function

When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle. We need to make several considerations when the equation involves trigonometric functions other than sine and cosine. Problems involving the reciprocals of the primary trigonometric functions need to be viewed from an algebraic perspective. In other words, we will write the reciprocal function, and solve for the angles using the function. Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function. First, as we know, the period of tangent is $\pi$, not $2\pi$. Further, the domain of tangent is all real numbers with the exception of odd integer multiples of $\tfrac{\pi}{2}$, unless, of course, a problem places its own restrictions on the domain.

Example. Solve the problem exactly: $2\sin^2\theta-1=0,\ 0\le\theta<2\pi$.

Solution. As this problem is not easily factored, we will solve using the square root property. First, we use algebra to isolate $\sin\theta$. Then we will find the angles.

Example. Solve the following equation exactly: $\csc\theta=-2,\ 0\le\theta<4\pi$.

Solution. We want all values of $\theta$ for which $\csc\theta=-2$ over the interval $0\le\theta<4\pi$.

Analysis. As $\sin\theta=-\tfrac12$, notice that all four solutions are in the third and fourth quadrants.

Example. Solve the equation exactly: $\tan\left(\theta-\tfrac{\pi}{2}\right)=1,\ 0\le\theta<2\pi$.

Solution. Recall that the tangent function has a period of $\pi$. On the interval $[0,\pi)$, and at the angle of $\tfrac{\pi}{4}$, the tangent has a value of $1$. However, the angle we want is $\left(\theta-\tfrac{\pi}{2}\right)$. Thus, if $\tan\left(\tfrac{\pi}{4}\right)=1$, then

Over the interval $[0,2\pi)$, we have two solutions:

Example. Identify all exact solutions to the equation $2(\tan x+3)=5+\tan x,\ 0\le x<2\pi$.

Solution. We can solve this equation using only algebra. Isolate the expression $\tan x$ on the left side of the equals sign.

There are two angles on the unit circle that have a tangent value of $-1$: $\theta=\tfrac{3\pi}{4}$ and $\theta=\tfrac{7\pi}{4}$.

Solve Trigonometric Equations Using a Calculator

Not all functions can be solved exactly using only the unit circle. When we must solve an equation involving an angle other than one of the special angles, we will need to use a calculator. Make sure it is set to the proper mode, either degrees or radians, depending on the criteria of the given problem.

Example. Use a calculator to solve the equation $\sin\theta=0.8$, where $\theta$ is in radians.

Solution. Make sure mode is set to radians. To find $\theta$, use the inverse sine function. On most calculators, you will need to push the 2ND button and then the SIN button to bring up the $\sin^{-1}$ function. What is shown on the screen is $\sin^{-1}($. The calculator is ready for the input within the parentheses. For this problem, we enter $\sin^{-1}(0.8)$, and press ENTER. Thus, to four decimal places,

The solution is

The angle measurement in degrees is

Analysis. Note that a calculator will only return an angle in quadrants I or IV for the sine function, since that is the range of the inverse sine. The other angle is obtained by using $\pi-\theta$.

Example. Use a calculator to solve the equation $\sec\theta=-4$, giving your answer in radians.

Solution. We can begin with some algebra.

Check that the MODE is in radians. Now use the inverse cosine function.

Since $\tfrac{\pi}{2}\approx1.57$ and $\pi\approx3.14$, $1.8235$ is between these two numbers, thus $\theta\approx1.8235$ is in quadrant II. Cosine is also negative in quadrant III. Note that a calculator will only return an angle in quadrants I or II for the cosine function, since that is the range of the inverse cosine, as shown below.

So, we also need to find the measure of the angle in quadrant III. In quadrant III, the reference angle is $\theta'\approx\pi-1.8235\approx1.3181$. The other solution in quadrant III is $\pi+1.3181\approx4.4597$.

The solutions are $1.8235\pm2\pi k$ and $4.4597\pm2\pi k$.

Solving Trigonometric Equations in Quadratic Form

Solving a quadratic equation may be more complicated, but once again, we can use algebra as we would for any quadratic equation. Look at the pattern of the equation. Is there more than one trigonometric function in the equation, or is there only one? Which trigonometric function is squared? If there is only one function represented and one of the terms is squared, think about the standard form of a quadratic. Replace the trigonometric function with a variable such as $x$ or $u$. If substitution makes the equation look like a quadratic equation, then we can use the same methods for solving quadratics to solve the trigonometric equations.

Example. Solve the equation exactly: $\cos^2\theta+3\cos\theta-1=0,\ 0\le\theta<2\pi$.

Solution. We begin by using substitution and replacing $\cos\theta$ with $x$. It is not necessary to use substitution, but it may make the problem easier to solve visually. Let $\cos\theta=x$. We have

The equation cannot be factored, so we will use the quadratic formula $x=\tfrac{-b\pm\sqrt{b^2-4ac}}{2a}$.

Replace $x$ with $\cos\theta$, and solve. Thus,

Note that only the $+$ sign is used. This is because we get an error when we solve $\theta=\cos^{-1}\left(\tfrac{-3-\sqrt{13}}{2}\right)$ on a calculator, since the domain of the inverse cosine function is $[-1,1]$. However, there is a second solution:

This terminal side of the angle lies in quadrant I. Since cosine is also positive in quadrant IV, the second solution is

Example. Solve the equation exactly: $2\sin^2\theta-5\sin\theta+3=0,\ 0\le\theta\le2\pi$.

Solution. Using grouping, this quadratic can be factored. Either make the real substitution, $\sin\theta=u$, or imagine it, as we factor:

Now set each factor equal to zero.

Next solve for $\theta$: $\sin\theta\ne\tfrac32$, as the range of the sine function is $[-1,1]$. However, $\sin\theta=1$, giving the solution $\tfrac{\pi}{2}$.

Analysis. Make sure to check all solutions on the given domain as some factors have no solution.

Example. Solve exactly:

Solution. This problem should appear familiar as it is similar to a quadratic. Let $\sin\theta=x$. The equation becomes $2x^2+x=0$. We begin by factoring:

Set each factor equal to zero.

Then, substitute back into the equation the original expression $\sin\theta$ for $x$. Thus,

The solutions within the domain $0\le\theta<2\pi$ are $0,\pi,\tfrac{7\pi}{6},\tfrac{11\pi}{6}$.

If we prefer not to substitute, we can solve the equation by following the same pattern of factoring and setting each factor equal to zero.

Analysis. We can see the solutions on the graph below. On the interval $0\le\theta<2\pi$, the graph crosses the $x$-axis four times, at the solutions noted. Notice that trigonometric equations that are in quadratic form can yield up to four solutions instead of the expected two that are found with quadratic equations. In this example, each solution (angle) corresponding to a positive sine value will yield two angles that would result in that value.

We can verify the solutions on the unit circle as well.

Example. Solve the equation quadratic in form exactly: $2\sin^2\theta-3\sin\theta+1=0,\ 0\le\theta<2\pi$.

Solution. We can factor using grouping. Solution values of $\theta$ can be found on the unit circle:

Solving Trigonometric Equations Using Fundamental Identities

While algebra can be used to solve a number of trigonometric equations, we can also use the fundamental identities because they make solving equations simpler. Remember that the techniques we use for solving are not the same as those for verifying identities. The basic rules of algebra apply here, as opposed to rewriting one side of the identity to match the other side. In the next example, we use two identities to simplify the equation.

Example. Use identities to solve exactly the trigonometric equation over the interval $0\le x<2\pi$.

Solution. Notice that the left side of the equation is the difference formula for cosine.

From the unit circle, we see that $\cos x=\tfrac{\sqrt3}{2}$ when $x=\tfrac{\pi}{6},\tfrac{11\pi}{6}$.

Example. Solve the equation exactly using a double-angle formula: $\cos(2\theta)=\cos\theta$.

Solution. We have three choices of expressions to substitute for the double-angle of cosine. As it is simpler to solve for one trigonometric function at a time, we will choose the double-angle identity involving only cosine:

So, if $\cos\theta=-\tfrac12$, then $\theta=\tfrac{2\pi}{3}\pm2\pi k$ and $\theta=\tfrac{4\pi}{3}\pm2\pi k$; if $\cos\theta=1$, then $\theta=0\pm2\pi k$.

Example. Solve the equation exactly using an identity: $3\cos\theta+3=2\sin^2\theta,\ 0\le\theta<2\pi$.

Solution. If we rewrite the right side, we can write the equation in terms of cosine:

Our solutions are $\tfrac{2\pi}{3},\tfrac{4\pi}{3},\pi$.

Solving Trigonometric Equations with Multiple Angles

Sometimes it is not possible to solve a trigonometric equation with identities that have a multiple angle, such as $\sin(2x)$ or $\cos(3x)$. When confronted with these equations, recall that $y=\sin(2x)$ is a horizontal compression by a factor of $2$ of the function $y=\sin x$. On an interval of $2\pi$, we can graph two periods of $y=\sin(2x)$, as opposed to one cycle of $y=\sin x$. This compression of the graph leads us to believe there may be twice as many $x$-intercepts or solutions to $\sin(2x)=0$ compared to $\sin x=0$. This information will help us solve the equation.

Example. Solve exactly: $\cos(2x)=\tfrac12$ on $[0,2\pi)$.

Solution. We can see that this equation is the standard equation with a multiple of an angle. If $\cos(\alpha)=\tfrac12$, we know $\alpha$ is in quadrants I and IV. While $\theta=\cos^{-1}\tfrac12$ will only yield solutions in quadrants I and II, we recognize that the solutions to the equation $\cos\theta=\tfrac12$ will be in quadrants I and IV.

Therefore, the possible angles are $\theta=\tfrac{\pi}{3}$ and $\theta=\tfrac{5\pi}{3}$. So, $2x=\tfrac{\pi}{3}$ or $2x=\tfrac{5\pi}{3}$, which means that $x=\tfrac{\pi}{6}$ or $x=\tfrac{5\pi}{6}$. Does this make sense? Yes, because $\cos\left(2\left(\tfrac{\pi}{6}\right)\right)=\cos\left(\tfrac{\pi}{3}\right)=\tfrac12$.

Are there any other possible answers? Let us return to our first step.

In quadrant I, $2x=\tfrac{\pi}{3}$, so $x=\tfrac{\pi}{6}$ as noted. Let us revolve around the circle again:

so $x=\tfrac{7\pi}{6}$.

One more rotation yields

$x=\tfrac{13\pi}{6}>2\pi$, so this value for $x$ is larger than $2\pi$, so it is not a solution on $[0,2\pi)$.

In quadrant IV, $2x=\tfrac{5\pi}{3}$, so $x=\tfrac{5\pi}{6}$ as noted. Let us revolve around the circle again:

so $x=\tfrac{11\pi}{6}$.

One more rotation yields

$x=\tfrac{17\pi}{6}>2\pi$, so this value for $x$ is larger than $2\pi$, so it is not a solution on $[0,2\pi)$.

Our solutions are $\tfrac{\pi}{6},\tfrac{5\pi}{6},\tfrac{7\pi}{6},\text{and}\ \tfrac{11\pi}{6}$. Note that whenever we solve a problem in the form of $\sin(nx)=c$, we must go around the unit circle $n$ times.

Solving Right Triangle Problems

We can now use all of the methods we have learned to solve problems that involve applying the properties of right triangles and the Pythagorean Theorem. We begin with the familiar Pythagorean Theorem, $a^2+b^2=c^2$, and model an equation to fit a situation.

Example. Use the Pythagorean Theorem, and the properties of right triangles to model an equation that fits the problem.

One of the cables that anchors the center of the London Eye Ferris wheel to the ground must be replaced. The center of the Ferris wheel is $69.5$ meters above the ground, and the second anchor on the ground is $23$ meters from the base of the Ferris wheel. Approximately how long is the cable, and what is the angle of elevation (from ground up to the center of the Ferris wheel)? See the figure below.

Solution. Using the information given, we can draw a right triangle. We can find the length of the cable with the Pythagorean Theorem.

The angle of elevation is $\theta$, formed by the second anchor on the ground and the cable reaching to the center of the wheel. We can use the tangent function to find its measure. Round to two decimal places.

The angle of elevation is approximately $71.7^\circ$, and the length of the cable is $73.2$ meters.

Example. Use the Pythagorean Theorem, and the properties of right triangles to model an equation that fits the problem.

OSHA safety regulations require that the base of a ladder be placed $1$ foot from the wall for every $4$ feet of ladder length. Find the angle that a ladder of any length forms with the ground and the height at which the ladder touches the wall.

Solution. For any length of ladder, the base needs to be a distance from the wall equal to one fourth of the ladder’s length. Equivalently, if the base of the ladder is “$a$” feet from the wall, the length of the ladder will be $4a$ feet. See the figure below.

The side adjacent to $\theta$ is $a$ and the hypotenuse is $4a$. Thus,

The elevation of the ladder forms an angle of $75.5^\circ$ with the ground. The height at which the ladder touches the wall can be found using the Pythagorean Theorem:

Thus, the ladder touches the wall at $\sqrt{15}\,a$ feet from the ground.

Key concepts

• When solving linear trigonometric equations, we can use algebraic techniques just as we do solving algebraic equations. Look for patterns, like the difference of squares, quadratic form, or an expression that lends itself well to substitution.
• Equations involving a single trigonometric function can be solved or verified using the unit circle.
• We can also solve trigonometric equations using a graphing calculator.
• Many equations appear quadratic in form. We can use substitution to make the equation appear simpler, and then use the same techniques we use solving an algebraic quadratic: factoring, the quadratic formula, etc.
• We can also use the identities to solve trigonometric equations.
• We can use substitution to solve a multiple-angle trigonometric equation, which is a compression of a standard trigonometric function. We will need to take the compression into account and verify that we have found all solutions on the given interval.
• Real-world scenarios can be modeled and solved using the Pythagorean Theorem and trigonometric functions.

Practice

Solve linear trigonometric equations in sine and cosine

Solve equations involving a single trigonometric function

Solve trigonometric equations using a calculator

Solve trigonometric equations that are quadratic in form

Solve trigonometric equations using fundamental identities

Solve trigonometric equations with multiple angles

Solve right triangle problems

This section is adapted from Precalculus 2e, Section 7.5: Solving Trigonometric Equations by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the decorative photo of the Egyptian pyramids (Figure 1), which carries no mathematics beyond the credit line; recreated the four instructional figures as accessible spec-first SVGs — the reference-angle diagram accompanying the calculator secant example (two terminal rays in standard position with their arcs, sampled from the exact solved angles); the graph of $y=2\sin^2\theta+\sin\theta$ accompanying its Analysis, sampled as a dense polyline from the exact formula over $\left[-\tfrac{\pi}{6},\tfrac{13\pi}{6}\right]$ (never a freehand curve) with the source’s own $\tfrac{\pi}{6}$-multiple tick labels; the Ferris wheel right triangle with the wheel drawn as a circle centered at the cable’s top vertex; and the generic ladder right triangle. Three sentences that cited a “unit circle” figure appearing elsewhere in the chapter (a different module’s figure, not this section’s own) are reworded to refer to the unit circle generically rather than point at a specific figure number, since that figure is outside this page’s scope. Every retained Try It became a real fillin component, each carrying the answerForm the printed subject demands to block a retype of the equation itself: radians on every interval-restricted list and general solution, with answerMode="unordered" on every list of two or more members (member order carries no meaning, and the engine grades a swapped order and an equation-wrapped restatement the same). Every general-solution question additionally names $k$ as the integer parameter, states the rounding its own printed key uses, and pins the representative angle’s range, since the grader compares one keyed representative strictly — instructions the source’s bare “find all solutions” wording leaves implicit. The Try It following Example 12 (a quadratic-in-form equation with no interval stated in the source, unlike every other equation in its subsection) is stated on $[0,2\pi)$ to match the domain its own printed answer key assumes and the domain used by the surrounding worked examples. Adapted fourteen selected end-of-section exercises — two linear, two single-function, two calculator/general-solution, two quadratic-form (one the “no real solution” case recast as a multiplechoice, since a categorical non-existence claim has no free-response answer), one identity-based quadratic-form, one pure-identity, two multiple-angle, and two right-triangle word problems — into a closing Practice block, one group per objective, every item independently re-derived (including by running the arithmetic in Node) rather than read off the source key.