Unit 5 · Topic 3.4
Sine and Cosine Function Graphs
Unwrap unit-circle coordinates into graphs and use their repeating landmarks to describe sine and cosine over all real inputs.
Learning Goals
- Construct sine and cosine graphs from unit-circle coordinates.
- Plot one cycle using five quarter-period landmarks.
- State domain, range, period, zeros, and extrema.
- Describe increasing and decreasing intervals from a graph.
- Use symmetry and horizontal shifts to compare sine and cosine.
1. Unwrap Circular Motion
For a unit-circle point \(P=(\cos\theta,\sin\theta)\), let the angle \(\theta\) become the horizontal graph input.
The sine graph tracks vertical displacement; the cosine graph tracks horizontal displacement as the point rotates counterclockwise.
2. Track One Sine Cycle
| \(\theta\) | 0 | \(\pi/2\) | \(\pi\) | \(3\pi/2\) | \(2\pi\) |
|---|---|---|---|---|---|
| \(\sin\theta\) | 0 | 1 | 0 | -1 | 0 |
| Motion | Midline, rising | Maximum | Midline, falling | Minimum | Midline, rising |
Connect these landmarks with a smooth curve. The direction at each midline crossing is part of the graph's identity.
3. Track One Cosine Cycle
| \(\theta\) | 0 | \(\pi/2\) | \(\pi\) | \(3\pi/2\) | \(2\pi\) |
|---|---|---|---|---|---|
| \(\cos\theta\) | 1 | 0 | -1 | 0 | 1 |
| Motion | Maximum | Midline, falling | Minimum | Midline, rising | Maximum |
Sine begins at the midline and rises; cosine begins at a maximum. Both complete the same pattern length.
4. Read the Parent Wave
The displayed wave repeats after one full revolution. A correct graph keeps equal horizontal spacing between maximum, midline, minimum, midline, and the next maximum.
5. State Domain, Range, and Period
| Feature | \(y=\sin\theta\) | \(y=\cos\theta\) |
|---|---|---|
| Domain | All real numbers | All real numbers |
| Range | \([-1,1]\) | \([-1,1]\) |
| Period | \(2\pi\) | \(2\pi\) |
Angles may rotate without bound, while unit-circle coordinates never leave the interval from -1 to 1.
6. Locate Zeros
Sine is zero where the unit-circle point lies on the x-axis. Cosine is zero where it lies on the y-axis:
A zero is an input value. The graph point at a zero is \((\theta,0)\), not a point on the original unit circle.
7. Locate Maxima and Minima
| Function | Maximum 1 | Minimum -1 |
|---|---|---|
| Sine | \(\theta=\pi/2+2\pi k\) | \(\theta=3\pi/2+2\pi k\) |
| Cosine | \(\theta=2\pi k\) | \(\theta=\pi+2\pi k\) |
The output extrema are always 1 and -1; the listed inputs tell where they occur.
8. Describe Increasing and Decreasing Intervals
Over one convenient cycle, sine rises from its minimum to maximum and falls from maximum to minimum:
Cosine decreases on \((0,\pi)\) and increases on \((\pi,2\pi)\). Add \(2\pi k\) to repeat every interval.
9. Compare Symmetry
Reflecting the unit-circle point across the x-axis preserves x and reverses y:
Thus sine is odd and has rotational symmetry about the origin. Cosine is even and has reflective symmetry across the y-axis. A graph inconsistent with these tests cannot be the corresponding parent function.
10. Relate Sine and Cosine by a Shift
The two graphs have the same shape, offset by one quarter of a cycle:
For example, cosine's maximum at \(\theta=0\) corresponds to sine's maximum after its input advances by \(\pi/2\).
11. Construct a Graph Efficiently
- Mark an interval of width \(2\pi\).
- Divide it into four equal parts of width \(\pi/2\).
- Plot the five landmark outputs.
- Connect them smoothly with correct direction.
- Repeat the complete pattern left and right.
Quarter-period spacing is horizontal spacing. It must not be confused with output height.
12. Recover Missing Table Values
Suppose a sine table gives \(f(0)=0\), \(f(\pi/2)=1\), and \(f(\pi)=0\). Unit-circle motion and symmetry determine
Use exact values when inputs are standard angles; do not estimate values that the unit circle determines exactly.
13. Distinguish Sine from Cosine
A graph with range \([-1,1]\) and period \(2\pi\) could be either function. Inspect its value and direction at zero:
| \(y=\sin\theta\) | Passes through \((0,0)\) while increasing. |
|---|---|
| \(y=\cos\theta\) | Has a maximum at \((0,1)\). |
One feature alone may not identify a translated or reflected sinusoid, but it distinguishes the two parent graphs.
14. Use Technology as a Verification Tool
- Use radian mode.
- Set an x-window containing at least one full \(2\pi\) cycle.
- Use a y-window extending slightly beyond \([-1,1]\).
- Trace key points and compare them with exact unit-circle values.
A narrow or uneven graphing window can make the period appear incorrect. The window displays the function; it does not define its domain or range.
15. Common Errors
- Starting sine at a maximum instead of at the origin.
- Starting cosine at the origin instead of at \((0,1)\).
- Using period \(\pi\) instead of \(2\pi\).
- Spacing the five landmarks unequally.
- Confusing zeros with maxima or minima.
- Reporting visible-window bounds as the domain.
- Drawing sharp corners instead of a smooth parent wave.
Key Takeaways
- Read zeros, extrema, intervals, range, and periodicity from sine and cosine graphs.
- Core relationship: \(\sin(x+2\pi)=\sin x,\quad\cos(x+2\pi)=\cos x\)
- Error check: Do not infer amplitude from the period.
- Construct one sine cycle using five key points.
- State all zeros and maximum inputs of cosine.
- Give the increasing intervals of sine over \([-2\pi,2\pi]\).
- Explain why cosine is even using unit-circle coordinates.
- Identify whether a graph passing through \((0,0)\) and initially decreasing is a parent sine or cosine graph, and explain what transformation may be present.