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Find sin (-5 ) Find the values of ϴ for which cos ϴ =1 is true. Graph the function y = sin x for the interval. Warm up. 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. - PowerPoint PPT Presentation
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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>.