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Graphs of Trigonometric Functions

Periodic Functions

A periodic function is one in which the function values repeat at regular intervals. Generally, a periodic function is defined as follows:
For a function , if there exists a non-zero constant such that for any real number in the domain of ,
then the function is called a periodic function, and the smallest positive value of such is called the period of the function.
The graph of a periodic function repeats the same shape for each period.

Graph of the Sine Function

As shown in the diagram, let be the point where the terminal side of an angle intersects the unit circle. Since , the value of is determined by the -coordinate of point .
As point moves along the unit circle, plotting the ordered pairs on a coordinate plane produces the graph of the sine function , as shown below.
  • The domain of is the set of all real numbers, and the range is .
  • The graph is symmetric about the origin, so .
  • Since the graph repeats its shape every , for any real number , Thus, the function is a periodic function with a period of .
Since the variable representing the domain is typically written as , we rewrite the sine function as

Properties of the Function

  1. The domain is the set of all real numbers.
  2. The range is .
  3. The function is continuous for all real numbers.
  4. The graph is symmetric about the origin, meaning .
  5. Since for any integer , the function is periodic with period .

Graph of the Cosine Function

As shown in the diagram, let be the point where the terminal side of an angle intersects the unit circle. Since , the value of is determined by the -coordinate of point .
As point moves along the unit circle, plotting the ordered pairs on a coordinate plane produces the graph of the cosine function , as shown below.
  • The domain of is the set of all real numbers, and the range is .
  • The graph is symmetric about the -axis, so .
  • Since the graph repeats its shape every , for any real number , Thus, the function is a periodic function with a period of .

As with the sine function, we rewrite the cosine function as

Properties of the Function

  1. The domain is the set of all real numbers.
  2. The range is .
  3. The function is continuous for all real numbers.
  4. The graph is symmetric about the -axis, meaning .
  5. Since for any integer , the function is periodic with period .

Graph of the Tangent Function

As shown in the diagram, let be the point where the terminal side of an angle intersects the unit circle. Let be the point where the tangent line to the unit circle at intersects the terminal side of . Since ( ), the value of is determined by the -coordinate of point .
As point moves along the unit circle, plotting the ordered pairs on a coordinate plane produces the graph of the tangent function , as shown below.
  • When ( is an integer), the terminal side of lies on the -axis, so is undefined.
  • The domain of the function is the set of all real numbers except ( is an integer), and its range is the set of all real numbers.
  • The vertical lines ( is an integer) are the asymptotes of the graph of .
  • The graph of is symmetric about the origin, which means .
  • The graph repeats its shape every interval. For any real number , Thus, is a periodic function with a period of .
As before, we rewrite the function as

Properties of the Function

  1. The domain consists of all real numbers except (where is an integer).
  2. The range is the set of all real numbers.
  3. The graph of the function is symmetric about the origin, meaning .
  4. It is a periodic function with a period of .
  5. The vertical asymptotes of the graph are the lines (where is an integer).
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