1. Table of Contents 6. Graphs of Sin and Cos

2. Graphs of Sin and Cos
Essential Question – What do the graphs of sin and cos look like and what real-life situations are modeled by them?

3. When will I use this? Electricity EKG Pendulum Respiratory cycles
Tuning fork
Seismograph

4. Graph vocabulary Amplitude – how high and low the graph goes
think amps on a speaker
Period – how long the graph takes to repeat itself
think about periods in history or fads
Phase shift – horizontal shifts in the graph
Think about how you go through phases, you shift or change
Vertical shift – a shift up or down in the graph

5. Graph of sin Repeats itself Standard form = amplitude
y = A sin kθ
= amplitude
Period = 2π/k or 360/k
In one period, there are 4 equal intervals
Begins at (0,0)
Goes up to +A
Comes back to x-axis
Goes down to –A
Repeats itself

6. Graph of y = sin θ

7. Graph of cos Repeats itself Standard form = amplitude
y = A cos kθ
= amplitude
Period = 2π/k or 360/k
In one period, there are 4 equal intervals
Begins at (0,+A)
Goes down x-axis
Goes down to -A
Comes back to x-axis
Goes back to +A
Repeats itself

8. Graph of y = cos θ

9. Example: Sketch the graph of y = 3 cos θ
Amplitude = 3, Period = 2π
Break the interval into four equal parts. 2π/4 = π/2
max
x-int
min 3 -3
y = 3 cos θ 2  θ y θ
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)
Example: y = 3 cos x

10. Example: Sketch the graph of y = 2 sin 4θ
Amplitude = 2, Period = 2π/4 = π/2
Break the interval into four equal parts. π/2/4 = π/8
x-int
min
max -2 2
y = 2cos 4θ
/2
/4 θ y θ
Example: y = 3 cos x

11. Example: Sketch the graph of y = -2 sin 3θ.
period: 2 3 =
amplitude: |a| = |–2| = 2
Calculate the five key points.
interval: 2 6 = 2 –2
y = –2 sin 3θ θ y θ
( , 2)
(0, 0)
( , 0)
( , 0)
( , -2)
Example: y = 2 sin(-3x)

12. Example: Sketch the graph of y = -4 cos (3/2 θ)
Amplitude = 4, Period = 2π/(3/2) = 4π/3
Break the interval into four equal parts. 4π/3/4 = 4π/12= π/3
min
x-int
max -4 4
y=-4 cos 3/2θ
4/3
2/3 θ y θ
Example: y = 3 cos x

13. Example: Sketch the graph of y = sin (1/3 θ)
Amplitude = 1, Period = 2π/(1/3) = 6π
Break the interval into four equal parts. 6π/4 = 3π/2
Example: y = 3 cos x

14. Assessment
Pretend to write a letter to a friend and explain how to graph sin and cos. Use 1 or 2 examples to help explain.