# Periodic Functions

## Inverse Trigonometric Functions

For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we will explore the inverse trigonometric functions.

### Understanding and Using the Inverse Sine, Cosine, and Tangent Functions

In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa, as summarized in .
For example, if 
       then we would write 
       Be aware that 
       does not mean 
       The following examples illustrate the inverse trigonometric functions:
1. Since 
  then 
  
2. Since 
         then 
        
      
3. Since 
         then 
        
      
In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for a one-to-one function, if 
  then an inverse function would satisfy 
 

Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a domain for each function that includes the number 0.  shows the graph of the sine function limited to 
  and the graph of the cosine function limited to 
 

 shows the graph of the tangent function limited to 
 

These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote.
On these restricted domains, we can define the inverse trigonometric functions.
1. The inverse sine function 
  means 
  The inverse sine function is sometimes called the arcsine function, and notated 
 

    
2. The inverse cosine function 
  means 
  The inverse cosine function is sometimes called the arccosine function, and notated 
 

    
3. The inverse tangent function 
  means 
  The inverse tangent function is sometimes called the arctangent function, and notated 
 

    
The graphs of the inverse functions are shown in , , and . Notice that the output of each of these inverse functions is a number, an angle in radian measure. We see that 
  has domain 
  and range 
  
  has domain 
  and range 
  and 
  has domain of all real numbers and range 
  To find the domain and range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line 
 


### Finding the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions

Now that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must evaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other numerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically 
  (30°), 
  (45°), and 
  (60°), and their reflections into other quadrants.

### Using a Calculator to Evaluate Inverse Trigonometric Functions

To evaluate inverse trigonometric functions that do not involve the special angles discussed previously, we will need to use a calculator or other type of technology. Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. These may be labeled, for example, SIN
 
 , ARCSIN, or ASIN.
In the previous chapter, we worked with trigonometry on a right triangle to solve for the sides of a triangle given one side and an additional angle. Using the inverse trigonometric functions, we can solve for the angles of a right triangle given two sides, and we can use a calculator to find the values to several decimal places.
In these examples and exercises, the answers will be interpreted as angles and we will use 
  as the independent variable. The value displayed on the calculator may be in degrees or radians, so be sure to set the mode appropriate to the application.

### Finding Exact Values of Composite Functions with Inverse Trigonometric Functions

There are times when we need to compose a trigonometric function with an inverse trigonometric function. In these cases, we can usually find exact values for the resulting expressions without resorting to a calculator. Even when the input to the composite function is a variable or an expression, we can often find an expression for the output. To help sort out different cases, let 
    and 
    be two different trigonometric functions belonging to the set 
    and let 
    and 
    be their inverses.

### Evaluating Compositions of the Form f(f−1(y)) and f−1(f(x))

For any trigonometric function, 
     for all 
     in the proper domain for the given function. This follows from the definition of the inverse and from the fact that the range of 
     was defined to be identical to the domain of 
     However, we have to be a little more careful with expressions of the form 
    
   

### Evaluating Compositions of the Form f−1(g(x))

Now that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a trigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form 
     For special values of 
     we can exactly evaluate the inner function and then the outer, inverse function. However, we can find a more general approach by considering the relation between the two acute angles of a right triangle where one is 
     making the other 
     Consider the sine and cosine of each angle of the right triangle in .
Because 
     we have 
     if 
     If 
     is not in this domain, then we need to find another angle that has the same cosine as 
     and does belong to the restricted domain; we then subtract this angle from 
     Similarly, 
     so 
     if 
     These are just the function-cofunction relationships presented in another way.

### Evaluating Compositions of the Form f(g−1(x))

To evaluate compositions of the form 
     where 
     and 
     are any two of the functions sine, cosine, or tangent and 
     is any input in the domain of 
     we have exact formulas, such as 
     When we need to use them, we can derive these formulas by using the trigonometric relations between the angles and sides of a right triangle, together with the use of Pythagoras’s relation between the lengths of the sides. We can use the Pythagorean identity, 
     to solve for one when given the other. We can also use the inverse trigonometric functions to find compositions involving algebraic expressions.

### Key Concepts

1. An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
2. Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
3. For any trigonometric function 
     if 
     then 
     However, 
     only implies 
     if 
     is in the restricted domain of 
     See .
4. Special angles are the outputs of inverse trigonometric functions for special input values; for example, 
  See .
5. A calculator will return an angle within the restricted domain of the original trigonometric function. See .
6. Inverse functions allow us to find an angle when given two sides of a right triangle. See .
7. In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example, 
     See .
8. If the inside function is a trigonometric function, then the only possible combinations are 
     if 
     and 
     if 
      See  and .
9. When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See .
10. When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, evaluate the expressions.
For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.
For the following exercises, find the angle 
     in the given right triangle. Round answers to the nearest hundredth.
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.
For the following exercises, find the exact value of the expression in terms of 
    
    with the help of a reference triangle.

### Extensions

For the following exercises, evaluate the expression without using a calculator. Give the exact value.
For the following exercises, find the function if 
 


### Graphical


### Real-World Applications


### Chapter Review Exercises


### Graphs of the Sine and Cosine Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

### Graphs of the Other Trigonometric Functions

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period,
midline equation, and asymptotes.
For the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.
For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: 
  where the domain is the years since 1980 and the range is the population of the city.
For the following exercises, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

### Inverse Trigonometric Functions

For the following exercises, find the exact value without the aid of a calculator.

### Chapter Practice Test

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.
For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.
For the following exercises, find the amplitude, period, phase shift, and midline.
For the following exercises, find the period and horizontal shift of each function.
For the following exercises, graph the functions on the specified window and answer the questions.
For the following exercises, let 
 

For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.
For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.
For the following exercises, find the exact value.
For the following exercises, suppose 
 
Evaluate the following expressions.

For the following exercises, determine whether the equation is true or false.