Periodic Pulse Slides Periodic Pulse
Trigonometric Identities in Motion
Warm-up: The Rhythm of Nature
Look at the patterns below. Both sound waves and ocean tides follow a repeating cycle.
Discussion Questions:
What happens to the graph if we look at it 24 hours later?
How would you describe the "repeat" interval?
Can we predict exactly where the wave will be in the future?
SOUND WAVE
OCEAN TIDES
Periodic Identities Video
Embedded media
Watch For:
Horizontal shifts by multiples of \(2\pi\).
The Rule:
\(f(\theta) = f(\theta + 2\pi n)\)
Goal:
Understand why the graph looks identical.
Key Concept: The Period
Periodic Function
A function that repeats its values at regular intervals (periods). For Sine and Cosine, this interval is \(2\pi\) .
The Identity Formulas:
Sine & Cosine
\(\sin(x + 2\pi n) = \sin(x)\)
\(\cos(x + 2\pi n) = \cos(x)\)
Tangent
\(\tan(x + \pi n) = \tan(x)\)
Adding a multiple of the period is like spinning around a circle and ending up in the exact same spot.
Main Activity: The Shifting Wave
1
Predict
Look at your Function Cards. Predict if the shift will produce an identical graph to \(y = \sin(x)\).
2
Verify
Use your Graphing Paper to calculate values at \(x = 0\) and \(x = \pi/2\) for each shifted function.
3
Sketch
Sketch the base wave and the "new" wave to see if they overlap perfectly.
Shift Examples
\(y = \sin(x + 2\pi)\)
\(y = \sin(x - 4\pi)\)
\(y = \sin(x + \pi)\)
Closure Challenge
Application
"If a ferris wheel completes a full turn every 2 minutes , write a periodic identity for its height \(H(t)\)."
Think About:
What is the period \(P\) in this scenario?
The Formula:
\(H(t) = H(t + \text{?})\)
Function Cards Wave Shift Cards
Cut along the dotted lines. Use these cards for "The Shifting Wave" activity.
Card A
\(y = \sin(x + 2\pi)\)
Card B
\(y = \sin(x - 4\pi)\)
Card C
\(y = \sin(x + \pi)\)
Card D
\(y = \sin(x - 6\pi)\)
Card E
\(y = \sin(x + 3\pi)\)
Card F
\(y = \sin(x + 10\pi)\)
Card G
\(y = \sin(x - \pi/2)\)
Card H
\(y = \sin(x + 4\pi)\)
Note for students: Assume n is an integer in the identity \(f(x) = f(x + Pn)\).
Wave Tracker Worksheet Wave Tracker
Analyzing Periodic Shifts in \(y = \sin(x)\)
Name:
Date:
1
The Prediction Table
Card Shifted Function Prediction Value at \(x=0\) Value at \(x=\pi/2\) Conclusion A y = sin(x + 2π) B y = sin(x - 4π) C y = sin(x + π) D y = sin(x - 6π) E y = sin(x + 3π) F y = sin(x + 10π) G y = sin(x - π/2) H y = sin(x + 4π)
2
Visual Verification
Base Wave vs. Card ______
0 \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\)
1 0.5 0
Base Wave vs. Card ______
0 \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\)
1 0.5 0
Synthesis Question
Based on your conclusion in the table, what is the mathematical rule for a horizontal shift that results in an identical sine function?
Periodic Pulse Answer Key Answer Key
Periodic Pulse: The Shifting Wave
Teacher Resource
Part 1: Prediction Table
Card Function Value at \(x=0\) Value at \(x=\pi/2\) Conclusion A sin(x + 2π) 0 1 Identical B sin(x - 4π) 0 1 Identical C sin(x + π) 0 -1 Different D sin(x - 6π) 0 1 Identical E sin(x + 3π) 0 -1 Different F sin(x + 10π) 0 1 Identical G sin(x - π/2) -1 0 Different H sin(x + 4π) 0 1 Identical
Synthesis & Closure
Synthesis Question Answer
The mathematical rule is that the shift must be an even multiple of \(\pi\) (e.g., \(2\pi, 4\pi, 6\pi\)). Algebraically, this is written as shifting by \(2\pi n\), where \(n\) is any integer.
Closure Challenge Answer
"If a ferris wheel completes a full turn every 2 minutes, write a periodic identity for its height \(H(t)\)."
\(H(t) = H(t + 2n)\)
Explanation: Since the period is 2 minutes, the height repeats exactly every 2 minutes. Shifting the time by 2 minutes (or any multiple of 2) results in the same height.
Teaching Tips
Common Misconception: Students often think any shift by \(\pi\) is periodic. Point out Card C and E to show that shifting by an odd multiple of \(\pi\) results in a reflection (negative sine).Unit Circle Connection: Remind students that \(2\pi\) is a full rotation, returning to the starting position.Visual Cues: If using the slides, pause at the animation in the video (0:49-2:25) to show the wave sliding over itself. Trig Graphing Paper Trig Graphing Paper
Standard Coordinate Planes for Sine & Cosine
Unit: Periodic Functions
Graph 1
0 \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(5\pi/2\) \(3\pi\) \(7\pi/2\) \(4\pi\)
1 0 -1
Graph 2
0 \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(5\pi/2\) \(3\pi\) \(7\pi/2\) \(4\pi\)
1 0 -1
Graph 3
0 \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(5\pi/2\) \(3\pi\) \(7\pi/2\) \(4\pi\)
1 0 -1
Graph 4
0 \(\pi/2\) \(\pi\) \(3\pi/2\) \(2\pi\) \(5\pi/2\) \(3\pi\) \(7\pi/2\) \(4\pi\)
1 0 -1
Each minor horizontal grid line represents an interval of \(\pi/2\).
