# Trigonometric Functions

## The Other Trigonometric Functions

A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is 
   or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions.

###  Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent 

To define the remaining functions, we will once again draw a unit circle with a point 
    corresponding to an angle of 
    as shown in . As with the sine and cosine, we can use the 
    coordinates to find the other functions.
The first function we will define is the tangent. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. In , the tangent of angle 
    is equal to 
    Because the y-value is equal to the sine of 
    and the x-value is equal to the cosine of 
    the tangent of angle 
    can also be defined as 
  The tangent function is abbreviated as 
    The remaining three functions can all be expressed as reciprocals of functions we have already defined.
1. The secant function is the reciprocal of the cosine function. In , the secant of angle 
     is equal to 
     The secant function is abbreviated as 
    
   
2. The cotangent function is the reciprocal of the tangent function. In , the cotangent of angle 
     is equal to 
     The cotangent function is abbreviated as 
    
   
3. The cosecant function is the reciprocal of the sine function. In , the cosecant of angle 
     is equal to 
     The cosecant function is abbreviated as 
    
   
Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting 
    equal to the cosine and 
    equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in .

### Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent

We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant.  shows which functions are positive in which quadrant.
To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “A,” ll of the six trigonometric functions are positive. In quadrant II, “Smart,” only ine and its reciprocal function, cosecant, are positive.   In quadrant III, “Trig,” only angent and its reciprocal function, cotangent, are positive.  Finally, in quadrant IV, “Class,” only osine and its reciprocal function, secant, are positive.

### Using Even and Odd Trigonometric Functions

To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.
Consider the function 
    shown in . The graph of the function is symmetrical about the y-axis. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation: 
    
   
   and so on. So 
     is an even function, a function such that two inputs that are opposites have the same output. That means 
    
  
Now consider the function 
    shown in . The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So 
    is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means 
    
We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in . The sine of the positive angle is 
    The sine of the negative angle is −y. The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in .

###  Recognizing and Using Fundamental Identities

We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.

### Alternate Forms of the Pythagorean Identity

We can use these fundamental identities to derive alternative forms of the Pythagorean Identity, 
     One form is obtained by dividing both sides by 
    
    
The other form is obtained by dividing both sides by 
    
   
As we discussed in the chapter opening, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or 
    
    will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.
Other functions can also be periodic. For example, the lengths of months repeat every four years. If 
     represents the length time, measured in years, and 
     represents the number of days in February, then 
    
    This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.

### Evaluating Trigonometric Functions with a Calculator

We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.
 Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.
If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor 
  to convert the degrees to radians. To find the secant of 
  we could press
or

### Key Equations


### Key Concepts

1. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle.
2. The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
3. The six trigonometric functions can be found from a point on the unit circle. See .
4. Trigonometric functions can also be found from an angle. See  .
5. Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See .
6. A function is said to be even if 
   and odd if 
  
 
7. Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
8. Even and odd properties can be used to evaluate trigonometric functions. See .
9. The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
10. Identities can be used to evaluate trigonometric functions. See  and . 
11. Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See .
12. The trigonometric functions repeat at regular intervals.
13. The period 
   of a repeating function 
   is the smallest interval such that 
   for any value of 
  
 
14. The values of trigonometric functions of special angles can be found by mathematical analysis.
15. To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the exact value of each expression.
For the following exercises, use reference angles to evaluate the expression.

### Graphical

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

### Technology

For the following exercises, use a graphing calculator to evaluate.

### Extensions

For the following exercises, use identities to evaluate the expression.
For the following exercises, use identities to simplify the expression.

### Real-World Applications
