# Trigonometric Identities and Equations

## 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. 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 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 ). 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.

### 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.

### 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 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.

### 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 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.

### Key Concepts

1. 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. See , , and .
2. Equations involving a single trigonometric function can be solved or verified using the unit circle. See , , and , and .
3. We can also solve trigonometric equations using a graphing calculator. See  and .
4. 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. See , , , and .
5. We can also use the identities to solve trigonometric equation. See , , and .
6. 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. See .
7. Real-world scenarios can be modeled and solved using the Pythagorean Theorem and trigonometric functions. See .

### Section Exercises


### Verbal


### Algebraic

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.

### Graphical

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.

### Technology

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.

### Extensions

For the following exercises, find all solutions exactly to the equations on the interval 
 


### Real-World Applications

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.

### Chapter Review Exercises


### Solving Trigonometric Equations with Identities

For the following exercises, find all solutions exactly that exist on the interval 
    
   
For the following exercises, use basic identities to simplify the expression.
For the following exercises, determine if the given identities are equivalent.

### Sum and Difference Identities

For the following exercises, find the exact value.
For the following exercises, prove the identity.
For the following exercise, simplify the expression.
For the following exercises, find the exact value.

### Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.
For the following exercises, use  to find the desired quantities.
For the following exercises, prove the identity.
For the following exercises, rewrite the expression with no powers.

### Sum-to-Product and Product-to-Sum Formulas

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.
For the following exercises, evaluate the sum by using a product formula. Write the exact answer.
For the following exercises, change the functions from a product to a sum or a sum to a product.

### Solving Trigonometric Equations

For the following exercises, find all exact solutions on the interval 
    
   
For the following exercises, find all exact solutions on the interval 
    
   
For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval 
     Round to four decimal places.
For the following exercises, graph each side of the equation to find the approximate solutions on the interval 
    
   

### Practice Test

For the following exercises, simplify the given expression.
For the following exercises, find the exact value.
For the following exercises, simplify each expression. Do not evaluate.
For the following exercises, find all exact solutions to the equation on 
   
  
For the following exercise, rewrite the product as a sum or difference.
For the following exercise, rewrite the sum or difference as a product.
For the following exercises, find all solutions exactly on the interval 
      
     
For the following exercises, prove the identity.