1. How does the number in front effect the graph?
Graph on GUT
y1 = sinx
y2 = 2 sinx
y3 = 3 sinx
y4 = ½ sinx
How does the number in front effect the graph?
Trigonometry

2. Amplitude and period of sine and cosine functions
6.4
Amplitude and period of sine and cosine functions
Trigonometry

3. Amplitude
Amplitude is defined to be ½ the distance between the lowest and highest points on the graph.
The “amplitude” of y = A sin x is |A|
Because it is defined to be a distance amplitude is always positive.
Trigonometry

4. Amplitude examples: Y = 3 sin x Y = -2 sinx Y = 1/3 cos x
What does the negative do to the graph? Reflects over x axis.
Trigonometry

5. How does the number in front of x effect the graph?
Graph on GUT
y1 = sin x
y2 = sin 2x
y3 = sin 3x
y4 = sin ½x
How does the number in front of x effect the graph?
Trigonometry

6. Period of a function
The period is the distance it takes for a graph to “do its thing.”
Period of a y = A sin x or y = A cos x is 2π.
The period of y = A sin kx or y = A cos kx is 2π/k.
If the p < 2π then graph is squished horizontally.
If the p > 2π then the graph is stretched horizontally.
Trigonometry

7. State the amplitude and period for each.
f(x) = 4 cos x
f(x) = -2 sin ½ x
f(x) = 1/3 cos 2x
A = 4
p = 2π
A = 2
p = 4π
A = 1/3
p = π
Trigonometry

8. Basic Forms
Y = ± A sin (kx)
Y = ± A cos (kx)
Trigonometry

9. Graph each. Y = -3 sin (x/4), -4π < x < 8π A = |-3| A = 3
P = 2π/k
P = 2π/(1/4)
P = 8π
Trigonometry

10. Graph each. 2. Y = -2 cos (x/2), -4π < x < 8π A = |-2| A = 2
P = 2π/k
P = 2π/(1/2)
P = 4π
Trigonometry

11. Graph each. 3. Y = 1/2 sin (4x), -π < x < π A = |1/2| A = 1/2
Trigonometry

12. Graph. 4. y = 4 sin x, -2π < x < 2π A = |4| A = 4 P = 2π/k
Trigonometry

13. Graph. 5. y = 3 cos (x/4), -π < x < π A = |3| A = 3 P = 2π/k
Trigonometry

14. Graph 6. Y = 1/3 cos (4x), -π < x < π A = |1/3| A = 1/3 P = 2π/k
Trigonometry

15. Find the amplitude and period for each function.
1. A piano tuner strikes a tuning fork for note A above middle C and sets in motion vibrations can by modeled by the equation y = sin 880π t. 2. A buoy that bobs up and down in the waves can be modeled by y= 1.75 cos π/3 t. 3. A pendulum can be modeled by the function d= 4 cos 8π t, where d is the horizontal displacement and t is time.
Trigonometry

16. Y = ± A cos (kx) |A| = 9.8 A = ±9.8 p = 6π 2π/k = 6π 2π = 6πk ⅓ = k
Write the equation of the cosine function with amplitude 9.8 and period 6π.
Y = ± A cos (kx)
|A| = 9.8
A = ±9.8
p = 6π
2π/k = 6π
2π = 6πk
⅓ = k
Y = ± 9.8 cos ⅓ x or y = ± 9.8 cos x/3
Trigonometry

17. Y = ± A sin (kx) |A| = 4.1 A = ±4.1 p = π/2 2π/k = π/2 4π = πk 4 = k
Write the equation of the sine function with amplitude 4.1 and period π/2.
Y = ± A sin (kx)
|A| = 4.1
A = ±4.1
p = π/2
2π/k = π/2
4π = πk
4 = k
Y = ± 4.1 sin 4x
Trigonometry

18. Y = ± A cos (kx) |A| = 2 A = ±2 p = π/2 2π/k = π/2 4π = πk 4 = k
Write the equation of the cosine function with amplitude 2 and period π/2.
Y = ± A cos (kx)
|A| = 2
A = ±2
p = π/2
2π/k = π/2
4π = πk
4 = k
Y = ± 2 cos 4x
Trigonometry

19. Y = ± A sin (kx) |A| = 0.5 A = ±0.5 p = 0.2π 2π/k = .2π 2π = .2πk
Write the equation of the sine function with amplitude 0.5 and period 0.2π.
Y = ± A sin (kx)
|A| = 0.5
A = ±0.5
p = 0.2π
2π/k = .2π
2π = .2πk
10 = k
Y = ± 0.5 sin 10x
Trigonometry

20. Y = ± A cos (kx) |A| = 1/5 A = ± 1/5 p = 2/5 π 2π/k = 2π/5 10π = 2πk
Write the equation of the cosine function with amplitude 1/5 and period 2/5 π.
Y = ± A cos (kx)
|A| = 1/5
A = ± 1/5
p = 2/5 π
2π/k = 2π/5
10π = 2πk
5 = k
Y = ± 1/5 cos 5x
Trigonometry

21. A = -3.5/2 (negative because it is on its way down) 2π/k = 14 2π = 14k
A signal buoy bobs up and down. From the highest point to the lowest point, the buoy moves a distance of 3.5 feet. It moves from its highest point down to its lowest point and back to its highest point every 14 seconds.
Write an equation of the motion for the buoy assuming that it is at its equilibrium point at t = 0 and the buoy is on its way down at that time.
Y = ± A sin kt
A = -3.5/2 (negative because it is on its way down)
2π/k = 14
2π = 14k
π/7 = k
y = sin π/7 t
Trigonometry

22. Determine the height of the buoy at 8 seconds and at 17 seconds
A signal buoy bobs up and down. From the highest point to the lowest point, the buoy moves a distance of 3.5 feet. It moves from its highest point down to its lowest point and back to its highest point every 14 seconds.
Determine the height of the buoy at 8 seconds and at 17 seconds
y = sin π/7 t
y = sin π/7 (8)
y ≈ 0.75
After 8 seconds, the buoy is about .8 feet above the equilibrium point.
y = sin π/7 (17)
y ≈ -1.71
After 17 seconds, the buoy is about 1.71 feet below the equilibrium point.
Trigonometry

23. A buoy bobs up and down. The distance between the highest and lowest point is 3 feet. It moves from its highest point down to its lowest point and back to its highest point every 8 seconds.
Find the equation of the motion for the buoy assuming that it is at its equilibrium point at t=0 and the buoy is on its way down up at that time. Y = ± A sin kt A = 3/2 (positive because it is on its way up) 2π/k = 8 2π = 8k π/4 = k y = 1.5 sin π/4 t
Trigonometry

24. A buoy bobs up and down. The distance between the highest and lowest point is 3 feet. It moves from its highest point down to its lowest point and back to its highest point every 8 seconds.
(b) Determine the height of the buoy at 3 seconds. y = 1.5 sin π/4 t y = 1.5 sin π/4 (3) y = 3.18 feet (c) Determine the height of the buoy at 12 seconds. y = 1.5 sin π/4 t y = 1.5 sin π/4 (12) y = feet
Trigonometry

25. A circuit has an alternating voltage of 100 volts that peaks every 0
A circuit has an alternating voltage of 100 volts that peaks every 0.5 seconds. Write a sin function for the voltage as a function of time t (in seconds).
Trigonometry

26. Frequency: the number of cycles per unit of time
Frequency = 1/period Period = 1/frequency hertz is a unit of frequency, One hertz = one cycle per second
Trigonometry

27. Write an equation of the sine function that represents the initial behavior of the vibrations of the note G above middle C having amplitude of and frequency of 392 hertz.
|A| = A = ± P = 1/frequency P = 1/392 P = 2π/k 1/392 = 2π/k K = 784π y = ±A sin kx A = ± sin 784πt
Trigonometry

28. 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.
|A| = 6 A = ± 6 P = 1/frequency P = 1/.1 P = 10 P = 2π/k 10 = 2π/k 10K = 2π k = π/5 y = ±A sin kx A = ± 6 sin π/5 t
Trigonometry

29. Exit Pass State the amplitude and period for f(x) = -2 sin (x/3).
Graph y = 2 cos (4x)
–π < x < π
Graph: y = -3 sin (x/2) π < x < 6π
Trigonometry

30. Even Answers
Trigonometry