1. y1 = sinx2. y2 = 2 sinx3. y3 = 3 sinx4. y4 = ½ sinx

How does the number in front effect the graph?

1Trigonometry

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.

3Trigonometry

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.

4Trigonometry

1. y1 = sin x2. y2 = sin 2x3. y3 = sin 3x4. y4 = sin ½x

How does the number in front of x effect the graph?

5Trigonometry

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.

6Trigonometry

1. f(x) = 4 cos x

2. f(x) = -2 sin ½ x

3. f(x) = 1/3 cos 2x

7Trigonometry

A = 4 p = 2π

A = 2 p = 4π

A = 1/3 p = π

Y = ± A sin (kx)

Y = ± A cos (kx)

8Trigonometry

1. Y = -3 sin (x/4), -4π < x < 8π

9Trigonometry

A = |-3| A = 3P = 2π/k P = 2π/(1/4)P = 8π

2. Y = -2 cos (x/2), -4π < x < 8π

10Trigonometry

A = |-2| A = 2P = 2π/k P = 2π/(1/2)P = 4π

3. Y = 1/2 sin (4x), -π < x < π

11Trigonometry

A = |1/2| A = 1/2P = 2π/4 P = π/2

4. y = 4 sin x, -2π < x < 2π

12Trigonometry

A = |4| A = 4P = 2π/k P = 2π/1P = 2π

5. y = 3 cos (x/4), -π < x < π

13Trigonometry

A = |3| A = 3P = 2π/kP = 2π/(1/4)P = 8π

6. Y = 1/3 cos (4x), -π < x < π

14Trigonometry

A = |1/3| A = 1/3P = 2π/kP = 2π/4P = π/2

15Trigonometry

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.

Y = ± A cos (kx) |A| = 9.8 A = ±9.8 p = 6π 2π/k = 6π 2π = 6πk ⅓ = kY = ± 9.8 cos ⅓ x or y = ± 9.8 cos x/3

16Trigonometry

Y = ± A sin (kx) |A| = 4.1 A = ±4.1 p = π/2 2π/k = π/2 4π = πk 4 = kY = ± 4.1 sin 4x

17Trigonometry

Y = ± A cos (kx) |A| = 2 A = ±2 p = π/2 2π/k = π/2 4π = πk 4 = kY = ± 2 cos 4x

18Trigonometry

Y = ± A sin (kx) |A| = 0.5 A = ±0.5 p = 0.2π 2π/k = .2π 2π = .2πk 10 = kY = ± 0.5 sin 10x

19Trigonometry

Y = ± A cos (kx) |A| = 1/5 A = ± 1/5 p = 2/5 π 2π/k = 2π/5 10π = 2πk 5 = kY = ± 1/5 cos 5x

20Trigonometry

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

21Trigonometry

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 ≈ -1.71After 17 seconds, the buoy is about 1.71 feet

below the equilibrium point. 22Trigonometry

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

23Trigonometry

(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

24Trigonometry

25Trigonometry

Frequency = 1/period

Period = 1/frequency

hertz is a unit of frequency,

One hertz = one cycle per second

26Trigonometry

|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

27Trigonometry

|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

28Trigonometry

1. State the amplitude and period for f(x) = -2 sin (x/3).

2. Graph y = 2 cos (4x) –π < x < π3. Graph: y = -3 sin (x/2)

-2π < x < 6π29Trigonometry

30Trigonometry