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Table of Contents 6. Graphs of Sin and Cos

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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.


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