1.
2.
3.
4.
y1 = sinx
y2 = 2 sinx
y3 = 3 sinx
y4 = ½ sinx
How does the number in front
effect the graph?
Trigonometry
1
6.4
Trigonometry

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
3
1.
Y = 3 sin x
1.
Y = -2 sinx
2.
Y = 1/3 cos x
3.
Y = - 3/2 cos x
What does the negative do to the graph? Reflects
over x axis.
Trigonometry
4
1.
2.
3.
4.
y1 = sin x
y2 = sin 2x
y3 = sin 3x
y4 = sin ½x
How does the number in front of
x effect the graph?
Trigonometry
5





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
6
1.
f(x) = 4 cos x
A=4
p = 2π
2.
f(x) = -2 sin ½ x
A=2
p = 4π
3.
f(x) = 1/3 cos 2x
Trigonometry
A = 1/3
p=π
7
Y
= ± A sin (kx)
Y
= ± A cos (kx)
Trigonometry
8
1.
Y = -3 sin (x/4), -4π < x < 8π
A = |-3|
A=3
P = 2π/k
P = 2π/(1/4)
P = 8π
Trigonometry
9
2. Y = -2 cos (x/2), -4π < x < 8π
A = |-2|
A=2
P = 2π/k
P = 2π/(1/2)
P = 4π
Trigonometry
10
3. Y = 1/2 sin (4x), -π < x < π
A = |1/2|
A = 1/2
P = 2π/4
P = π/2
Trigonometry
11
4. y = 4 sin x, -2π < x < 2π
A = |4|
A=4
P = 2π/k
P = 2π/1
P = 2π
Trigonometry
12
5. y = 3 cos (x/4), -π < x < π
A = |3|
A=3
P = 2π/k
P = 2π/(1/4)
P = 8π
Trigonometry
13
6. Y = 1/3 cos (4x), -π < x < π
A = |1/3|
A = 1/3
P = 2π/k
P = 2π/4
P = π/2
Trigonometry
14
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 = 0.001 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
15
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
16
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
17
Y = ± A cos (kx)
|A| = 2
A = ±2
p = π/2
2π/k = π/2
4π = πk
4=k
Y = ± 2 cos 4x

Trigonometry
18
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
19
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
20

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 = -1.75 sin π/7 t
Trigonometry
21
Determine the height of the buoy at 8 seconds and at
17 seconds
y = -1.75 sin π/7 t
y = -1.76 sin π/7 (8)
y ≈ 0.75
After 8 seconds, the buoy is about .8 feet above the
equilibrium point.
y = -1.75 sin π/7 t
y = -1.76 sin π/7 (17)
y ≈ -1.71
After 17 seconds, the buoy is about 1.71 feet below the
equilibrium point.

Trigonometry
22
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
23
(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 = 12.73 feet
Trigonometry
24
Trigonometry
25
Frequency = 1/period
Period = 1/frequency
hertz is a unit of frequency,
One hertz = one cycle per second
Trigonometry
26
|A| = 0.015
A = ± 0.015
P = 1/frequency
P = 1/392
P = 2π/k
1/392 = 2π/k
K = 784π
y = ±A sin kx
A = ± 0.015 sin 784πt
Trigonometry
27
|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
28
State the amplitude and period
for f(x) = -2 sin (x/3).
2. Graph y = 2 cos (4x)
–π < x < π
3. Graph: y = -3 sin (x/2)
2π < x < 6π
1.
Trigonometry
29
Trigonometry
30