The trigonometric functions cosine, sine, and tangent satisfy several properties of symmetry that are useful for understanding and evaluating these functions.
The cosine and sine functions satisfy the following properties of symmetry:
From the definition of cosine and sine in the unit circle,
We can see that for both and , remains the same. Thus, .
Similarly, we can see that the in two cases are additive inverse of each other. Thus,
Now that we have the above identities, we can prove several other identities, as shown in the following example.
Prove the identities
Using the properties of symmetry above, we can show that sine and cosine are special types of functions.
A function is an even function if and only if for all real values of , . In other words, the graph is symmetric about -axis.
A function is an odd function if and only if for all real values of , . In other words, the graph is symmetric about origin.
Also, . That is, an odd function must pass through the origin.
From this definition, the cosine function is an even function and the sine function is an odd function.
What symmetry is there between the angles and If we plug in a few values for , how do the basic trigonometric functions change?
By the properties of symmetry, we can write in terms of as follows:
Similarly, we can write in terms of as follows:
Determine whether the function
is an odd function, an even function, or neither.
The function satisfies
since is an even function. Therefore, is an even function.
Find a relationship between and
Solution 1: We have
Solution 2: We have
This shows the period of the tangent function is at most .
Since the cosine function satisfies , the graph of the function is symmetric about the -axis. What is the symmetry satisfied by the graph of
Since the function satisfies , the graph of is symmetric about the origin.
In general, for any even function , the the graph of is symmetric about the -axis; for any odd function , the graph of is symmetric about the origin.
See Sine and Cosine graphs for more properties of the sine and cosine graphs.