- 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.
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. He reasoned that when the height of his staff's shadow was exactly equal to the actual height of the staff, then the height of the nearby pyramid's shadow must also be equal to the height of the actual pyramid. Since the structures and their shadows were creating a right triangle with two equal sides, they were similar triangles. By measuring the length of the pyramid's shadow at that moment, he could obtain the height of the pyramid. 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 the 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 In other words, every units, the y-values repeat. If we need to find all possible solutions, then we must add where 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
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.
Solving a Linear Trigonometric Equation Involving the Cosine Function
Find all possible exact solutions for the equation
Solving a Linear Equation Involving the Sine Function
Find all possible exact solutions for the equation
Given a trigonometric equation, solve using algebra.
- Look for a pattern that suggests an algebraic property, such as the difference of squares or a factoring opportunity.
- Substitute the trigonometric expression with a single variable, such as or
- Solve the equation the same way an algebraic equation would be solved.
- Substitute the trigonometric expression back in for the variable in the resulting expressions.
- Solve for the angle.
Solve the Linear Trigonometric Equation
Solve the equation exactly:
Solve exactly the following linear equation on the interval
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 (see Figure 2). 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 not Further, the domain of tangent is all real numbers with the exception of odd integer multiples of unless, of course, a problem places its own restrictions on the domain.
Solving a Problem Involving a Single Trigonometric Function
Solve the problem exactly:
Solving a Trigonometric Equation Involving Cosecant
Solve the following equation exactly:
As notice that all four solutions are in the third and fourth quadrants.
Solving an Equation Involving Tangent
Solve the equation exactly:
Find all solutions for
Identify all Solutions to the Equation Involving Tangent
Identify all exact solutions to the equation
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.
Using a Calculator to Solve a Trigonometric Equation Involving Sine
Use a calculator to solve the equation where is in radians.
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 Thus, the additional solution is
Using a Calculator to Solve a Trigonometric Equation Involving Secant
Use a calculator to solve the equation giving your answer in radians.
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 or 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.
Solving a Trigonometric Equation in Quadratic Form
Solve the equation exactly:
Solving a Trigonometric Equation in Quadratic Form by Factoring
Solve the equation exactly:
Make sure to check all solutions on the given domain as some factors have no solution.
Solve [Hint: Make a substitution to express the equation only in terms of cosine.]
Solving a Trigonometric Equation Using Algebra
We can see the solutions on the graph in Figure 3. On the interval 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 in Figure 2 as well.
Solving a Trigonometric Equation Quadratic in Form
Solve the equation quadratic in form exactly:
Solve the quadratic equation
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.
Use Identities to Solve an Equation
Use identities to solve exactly the trigonometric equation over the interval
Solving the Equation Using a Double-Angle Formula
Solve the equation exactly using a double-angle formula:
Solving an Equation Using an Identity
Solve the equation exactly using an identity:
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 or When confronted with these equations, recall that is a horizontal compression by a factor of 2 of the function On an interval of we can graph two periods of as opposed to one cycle of This compression of the graph leads us to believe there may be twice as many x-intercepts or solutions to compared to This information will help us solve the equation.
Solving a Multiple Angle Trigonometric Equation
Solve exactly: on
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, and model an equation to fit a situation.
Using the Pythagorean Theorem to Model an Equation
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 Figure 4.
Using the Pythagorean Theorem to Model an Abstract 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.
Access these online resources for additional instruction and practice with solving trigonometric equations.
9.5 Section Exercises
Will there always be solutions to trigonometric function equations? If not, describe an equation that would not have a solution. Explain why or why not.
When solving a trigonometric equation involving more than one trig function, do we always want to try to rewrite the equation so it is expressed in terms of one trigonometric function? Why or why not?
When solving linear trig equations in terms of only sine or cosine, how do we know whether there will be solutions?
For the following exercises, find all solutions exactly on the interval
For the following exercises, solve exactly on
For the following exercises, find all exact solutions on
For the following exercises, solve with the methods shown in this section exactly on the interval
For the following exercises, solve exactly on the interval Use the quadratic formula if the equations do not factor.
For the following exercises, find exact solutions on the interval Look for opportunities to use trigonometric identities.
For the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the results by graphing the equation and finding the zeros.
For the following exercises, use a calculator to find all solutions to four decimal places.
For the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval Round to four decimal places.
For the following exercises, find all solutions exactly to the equations on the interval
An airplane has only enough gas to fly to a city 200 miles northeast of its current location. If the pilot knows that the city is 25 miles north, how many degrees north of east should the airplane fly?
If a loading ramp is placed next to a truck, at a height of 4 feet, and the ramp is 15 feet long, what angle does the ramp make with the ground?
If a loading ramp is placed next to a truck, at a height of 2 feet, and the ramp is 20 feet long, what angle does the ramp make with the ground?
A woman is watching a launched rocket currently 11 miles in altitude. If she is standing 4 miles from the launch pad, at what angle is she looking up from horizontal?
An astronaut is in a launched rocket currently 15 miles in altitude. If a man is standing 2 miles from the launch pad, at what angle is the astronaut looking down at him from horizontal? (Hint: this is called the angle of depression.)
A woman is standing 8 meters away from a 10-meter tall building. At what angle is she looking to the top of the building?
Issa is standing 10 meters away from a 6-meter tall building. Travis is at the top of the building looking down at Issa. At what angle is Travis looking at Issa?
A 20-foot tall building has a shadow that is 55 feet long. What is the angle of elevation of the sun?
A 90-foot tall building has a shadow that is 2 feet long. What is the angle of elevation of the sun?
A spotlight on the ground 3 meters from a 2-meter tall man casts a 6 meter shadow on a wall 6 meters from the man. At what angle is the light?
A spotlight on the ground 3 feet from a 5-foot tall woman casts a 15-foot tall shadow on a wall 6 feet from the woman. At what angle is the light?
For the following exercises, find a solution to the following word problem algebraically. Then use a calculator to verify the result. Round the answer to the nearest tenth of a degree.
A person does a handstand with their feet touching a wall and their hands 1.5 feet away from the wall. If the person is 6 feet tall, what angle do their feet make with the wall?
A person does a handstand with her feet touching a wall and her hands 3 feet away from the wall. If the person is 5 feet tall, what angle do her feet make with the wall?
A 23-foot ladder is positioned next to a house. If the ladder slips at 7 feet from the house when there is not enough traction, what angle should the ladder make with the ground to avoid slipping?