Warm up

Warm up

» Find sin (-5)» Find the values of ϴ for which cos ϴ=1 is

true.» Graph the function y = sin x for the

interval 1012 x

Lesson 6-4 Amplitude and Period of Sine and Cosine Functions

Objective: To find the amplitude and period of sine and cosine functions.To write equations of sine and cosine functions given the amplitude and

period.

Period

» In the last lesson we learned that the period was the domain over which the graph repeats.

» The period of and is where k > 0.

ky sin ky cosk

2

Amplitude

» For the equation y= A sin the maximum absolute value of the graph is 1.

» This is the amplitude.

Amplitude

» It can also be described as the absolute value of half the difference of the maximum and minimum values of the function.

» Example: y = 4sinθ The amplitude is 4…

The maximum is 4 and the minimum is -4. So (4 - -4)/2 = 4

Frequency

» Frequency is the number of cycles per unit of time.

» Period and Frequency are reciprocals of each other.

» Frequency is measured in units called hertz.» 1 hertz = 1 cycle per second

frequencyperiod

1

periodfrequency

1

Examples

» State the amplitude for the function

» State the amplitude and period for the function

then graph the function

sin2y

2cos5y

» Write an equation of the sine function with amplitude 2 and period

» Start with

» So

2

kAy sinky sin2

2

2

k4k

4sin2y

Practice

» A pendulum swings a total distance of 0.30 meter. The center point is zero. It completes a cycle every 2 seconds.

» Assuming that the pendulum is at the center point and heading right at t=0, find an equation for the motion of the pendulum.

sin15.y

Practice

» The Sears Building in Chicago sways back and forth at a vibration frequency of about 0.1 Hz. On average, it sways 6 inches from true center. Write an equation of the sine function that represents this behavior.

)2sin(.6 ty

interactive sine graph

Sources

» Interactive Mathematics. N.p., n.d. Web. 27 Oct. 2013. <http://www.intmath.com/ trigonometric-graphs/1-graphs-sine-cosine-amplitude.php>.

» Handbook for Acoustic Ecology. Cambridge Publishing, 1999. Web. 28 Oct. 2013. <http://www.sfu.ca/ sonic-studio/handbook/Sine_Wave.html>.