On the unit circle, which radian measure represents the point located at (− 1 2 , − 3 2 )? What about at ( 2 2 , − 2 2 )? Problem of the Day

Section 12-7 Graphing Trigonometric Functions

Then Now Objectives You examined periodic function. Describe and graph the sine, cosine, and tangent functions.

Common Core State Standards Content Standards F.IF.7.e – Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.TF.5 – Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Mathematical Practices 1) Make sense of problems and persevere in solving them. Common Core State Standards

Trigonometric functions can also be graphed on the coordinate plane Trigonometric functions can also be graphed on the coordinate plane. Remember that graphs of periodic functions have repeating patterns, or cycles. The horizontal length of each cycle is the period. The amplitude of the graph of a sine or cosine function equals half the difference between the maximum and minimum values of the function. Vocabulary

As with other functions, trigonometric functions can be transformed As with other functions, trigonometric functions can be transformed. For the graphs of y = a sin bθ and y = a cos bθ the amplitude = |a| and the period = 360° 𝑏 Amplitude and Period

Find the amplitude and period of each function: y = cos 1 2 𝜃 y = 3 sin 5𝜃 Example 1

Find the amplitude and period of each function: y = 4 sin 1 2 𝜃 y = 2 cos 3𝜃 Example 1

Trigonometric functions are useful when modeling real-world periodic motion such as electromagnetic waves or sound waves. Often these waves are described using frequency. Frequency is the number of cycles in a given unit of time. The frequency of the graph of a function is the reciprocal of the period of the function. So, if the period of a function is 1 100 second, then the frequency is 100 cycles per second. Vocabulary

Humans can hear sounds with frequencies as low as 20 hertz Humans can hear sounds with frequencies as low as 20 hertz. Find the period of the function. Let the amplitude equal 1 unit. Write a cosine equation to model the sound waves. Example 3

A seismic station detects an earthquake wave that has a frequency of ½ Hertz and an amplitude of 1 meter. Find the equation of the sine graph. Example 3

p. 841 #1. , 3. , 5, 9 – 19. odd (. Find the amplitude and period p.841 #1*, 3*, 5, 9 – 19* odd (*Find the amplitude and period. Do NOT graph.) Homework