5.2 – Trig Functions of Real Numbers
5.2 Video 1 of 2: Trigonometric Functions of Real Numbers.
5.2 Video 2 of 2: Pythagorean Identities.
Section Notes
Trigonometric Functions
A function is a relationship between two sets. It assigns an “output” for each “input”. For trigonometric functions, the “input” is the number , the distance traveled around the unit circle. The output is a value involving the and coordinates of the terminal point of , as defined below.
The Trigonometric Functions 

There are six trigonometric functions. For any real number with terminal point they are:

Find the values of the six trigonometric functions for 

First, note that the terminal point of is . Thus, the six trigonometric functions are:

Find the values of the six trigonometric functions for 

The terminal point of is , so the six trigonometric functions are:

Find the values of the six trigonometric functions for 

The terminal point of is , so the six trigonometric functions are:

Using the definitions of the trigonometric functions the chart below shows the six trigonometric functions for and :
We can make note of the numbers for which each function is undefined by defining the domain of each function. The domain of and is , all real numbers.
The domain of and is all real numbers except for , an integer.
The domain of and is all real numbers except for an integer.
The signs of each trigonometric function is determined by the quadrant for which lies.
In Quadrant I all trigonometric functions are positive
In Quadrant II and are positive. The other four are negative.
In Quadrant III and are positive. The other four are negative.
In Quadrant IV and are positive. The other four are negative.
Evaluate 

Because is in quadrant II, the value will be negative. The reference number for is which has coordinate , so then 
Evaluate 

is in quadrant I, so the value will be positive. The reference angle is and so we find that 
Evaluate 

, so the terminal point of is the same as the terminal point of , which lies in quadrant II, making positive. The reference angle for is , thus, 
While we have begun a list of trigonometric values for certain values, there are many times we will need to evaluate the trig functions for other values. For example, how would we estimate ? Engineers and mathematicians used to use books filled with tables to estimate such values. Slide rulers were also used to be able to estimate such values. Luckily, today, modern calculators have programs to evaluate such values. The calculator must be set to radian mode (as opposed to degrees). Inputting into a calculator and rounding to six decimal places gives 0.973848.
Use a calculator to find the value of . Round to six decimal places. 

Inputting this into a calculator yields 
Use a calculator to find the value of . Round to six decimal places. 

Inputting this into a calculator yields 
Use a calculator to find the value of . Round to six decimal places. 

Inputting this into a calculator yields 
Some calculators have the functions and and some do not. If your calculator does not have these functions, you can easily use the reciprocal relations to evaluate them:
Reciprocal relations 

Use a calculator to find the value of . Round to six decimal places. 

Use a calculator to find the value of . Round to six decimal places. 

When we view the relationship of the terminal point of and , as pictured below:
It can be seen that the terminal point of is , while the terminal point of is . Thus, the value of is the opposite of the value of . The is unchanged. This gives way to the even and odd properties of trigonometric functions:
EvenOdd Properties of Trig Functions 


Use the evenodd properties of the trigonometric functions to determine the value of 

is a n even function, so we have 
Use the evenodd properties of the trigonometric functions to determine the value of 

is an odd function, so we have 
It is also worth noting that for any terminal point, this point is on the unit circle, and thus, the and values must solve the equation of the unit circle. If we replace and with the expression and , respectively, we obtain the first of three Pythagorean Identities:
Pythagorean Identities 

These identities can often aid in finding the other trigonometric values when one of them is known
If and is in quadrant IV, find the values of all of the trigonometric functions at 

Using the first Pythagorean identity, we have
