The Wave Anatomy
8.1 -- Characteristics of sine and cosine graphs
Step 1 of 7: Unroll the Circle
We are going to track one thing: the y-coordinate, as the angle grows. Click "Next Point" to advance the angle around the unit circle. Watch where the dot lands on the graph to the right.
Click "Next Point" to place a dot on the circle and see its y-coordinate plotted on the graph.
Step 2 of 7: Fill in the Curve
Keep adding points. The more you add, the clearer the wave shape becomes. You are building the sine curve from the circle, point by point, by your own hand.
Add more points to fill in the wave shape. Watch how each point's height matches the y-coordinate on the circle.
Step 3 of 7: The Peak of the Curve
Pause at the top of the circle. What angle puts the point at the very top, where y is at its maximum?
Step 4 of 7: The Five Key Points
One full sine cycle has five key points. Look at the circle and the graph together. Which of these correctly lists all five key angle-value pairs?
Step 5 of 7: Now Track the x-Coordinate
This time we track the x-coordinate of the point on the circle. That gives us the cosine. Click "Next Point" and watch the blue curve build.
Click "Next Point" to track the x-coordinate (cosine) from the circle to the graph.
Step 6 of 7: Overlay Both Curves
Here are sine and cosine together. What do you notice about their relationship?
Step 7 of 7: Why These Starting Values?
Why does sine start at 0 and cosine start at 1?
Explore: Circle-to-Graph Tracing
Use the controls to trace the curves from the unit circle. Toggle sine and cosine on or off.
Try This
Starter: Read the Sine Graph
Look at the sine graph below. From the graph alone, identify the period, amplitude, midline, and all five key points of the first cycle.
What is the period of this sine curve?
What is the amplitude?
What is the midline?
Which list correctly gives the five key points?
Stretch: Galveston Tides
Below is simulated tide data from Galveston, Texas, measured over 48 hours. Tides rise and fall in a roughly sinusoidal pattern driven by the Moon's gravitational pull.
| Hour | 0 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 |
|---|---|---|---|---|---|---|---|---|---|
| Height (ft) | 1.8 | 2.7 | 1.8 | 0.9 | 1.8 | 2.7 | 1.8 | 0.9 | 1.8 |
| Hour | 27 | 30 | 33 | 36 | 39 | 42 | 45 | 48 |
|---|---|---|---|---|---|---|---|---|
| Height (ft) | 2.7 | 1.8 | 0.9 | 1.8 | 2.7 | 1.8 | 0.9 | 1.8 |
Does the tide data look more like sine or cosine?
Estimate the period in hours. (within 0.5 hours)
Estimate the amplitude in feet. (within 0.2 feet)
Why is the real tidal period not exactly 12 hours?
Challenge: Dallas Daylight Hours
In Dallas, the shortest day is in late December with about 10.1 hours of daylight. The longest day is in late June with about 14.3 hours. The daylight hours follow a roughly sinusoidal pattern over the year.
Estimate the amplitude in hours. (within 0.1 hours)
Hint: amplitude is half the distance from the minimum to the maximum.
Estimate the midline in hours. (within 0.1 hours)
Hint: the midline is the average of the maximum and minimum.
What is the period in days? (within 1 day)
On roughly what dates does the daylight curve cross the midline (about 12.2 hours), and what are those days called?
- Connect the unit circle to the graphs of sine and cosine by tracing coordinates.
- Identify the five key points of one cycle of sine and cosine.
- Define period, amplitude, and midline from a graph.
- Explain why cosine is a horizontal shift of sine (and vice versa).
- Explain why the period is \(2\pi\) and the amplitude is 1 for the standard sine and cosine functions.
Quick Check
A sine curve has a maximum at \(y = 1\) and a minimum at \(y = -1\). What is the amplitude?
Instructor Notes
Teaching Notes
This simulation builds the sine and cosine curves from the unit circle, making the connection concrete rather than abstract. The key pedagogical move is having students generate the curve point by point -- they see that every value on the graph is just a coordinate read off the circle.
The five key points framework (three zeros, one max, one min) gives students anchor points they can use to sketch any sinusoidal curve quickly by hand. Emphasize that these points correspond to the cardinal directions on the unit circle.
The overlay of sine and cosine in Step 6 is the right time to introduce the phase shift concept informally. Students see the same curve, offset. The formal identity \(\cos\theta = \sin(\theta + \frac{\pi}{2})\) comes later, but the visual intuition starts here.
Common Student Errors
- Amplitude = peak-to-trough: Students often compute 1 - (-1) = 2 and call the amplitude 2. This is the most common error. The simulation addresses it in the Quick Check directly.
- Sine and cosine are unrelated: Without the circle connection, students treat them as two separate functions to memorize. The overlay step makes the relationship visible.
- Period in degrees vs. radians: Students may say the period is 360 instead of \(2\pi\). Both are correct, but they need to know which unit is expected.
- Confusing starting values: "Sine starts at 1" is a common recall error. Returning to the circle at \(\theta = 0\) corrects it immediately.
Discussion Questions
- If the circle had radius 3 instead of 1, what would change about the sine curve?
- If you walked around the circle twice (0 to \(4\pi\)), what would the sine graph look like?
- Can you think of a real-world quantity that follows a sinusoidal pattern but with a period much shorter than \(2\pi\)? Much longer?
- Why do we call the radius-1 circle the "unit" circle, and why is it the default for defining sine and cosine?
Exam Connection
Exam questions on 8.1 typically ask students to identify period, amplitude, and midline from a graph, or to sketch one cycle given those parameters. The five key points method taught here maps directly to the "sketch by key points" technique most textbooks use. The Galveston tide problem in Apply Stretch mirrors exam word problems that ask students to model periodic data with a sinusoidal function.