Track the vertical coordinate for sine and horizontal coordinate for cosine on the unit circle. Their basic graphs repeat after one full revolution. Increasing/decreasing describes outputs; concavity describes how rates change. The unnumbered waves illustrate shape, not exact coordinates.
What a strong response shows
Sine follows vertical displacement; cosine follows horizontal displacement on the unit circle.
Use quarter-cycle landmarks to establish zeros, extrema and whether outputs rise or fall.
Describe concavity using how rates change, not whether the graph is above or below an axis.
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What You Need to Know
Sine records the vertical coordinate of a unit-circle point; cosine records its horizontal coordinate as the angle changes.
Track the unit-circle point in order. Use its vertical coordinate for sine and its horizontal coordinate for cosine.
Mark the quarter-turn positions. Plot their vertical coordinates for sine and their horizontal coordinates for cosine, then connect each set with its own smooth periodic curve.
Starting the sine graph at a peak or the cosine graph at zero.
What is the starting height of cosine at angle zero?
Learning Objectives
3.4.A — Construct representations of the sine and cosine functions using the unit circle.
Use the unit circle to anchor key points before drawing the full curve.
. Mark start, quarter-turn, half-turn, and full-turn. . Record vertical coordinates. . Connect smoothly. . Repeat the cycle.
Using equal horizontal spacing but unequal quarter-cycle order.
Which key point occurs halfway through one cycle?
Key Concepts
Both graphs have amplitude and period one full revolution. Sine starts on the midline; cosine starts at a maximum.
Use starting value and first direction to distinguish sine from cosine.
Sine moves upward from zero; cosine moves downward from its maximum.
Calling any wave that starts at zero cosine.
Which function starts at a maximum?
Relevant Representations
The unit circle and the wave graph are two views of the same changing coordinate.
| Angle stage | Sine height | Cosine height |
|---|---|---|
| start | midline | maximum |
| quarter turn | maximum | midline |
| half turn | midline | minimum |
| three-quarter turn | minimum | midline |
| full turn | midline | maximum |
Map angle to the graph input. The graph output is the circle point’s vertical coordinate for sine or horizontal coordinate for cosine.
Follow the circle point counterclockwise. At each marked angle, plot its vertical projection for sine and horizontal projection for cosine.
Using the x-coordinate for both sine and cosine graphs.
Which coordinate creates the sine height?
Essential Relationships or Formulas
The standard graphs repeat after one full turn and stay between their extrema.
Use anchors for shape; use identities for periodic checks.
Compare a graph’s start, peak, trough, and repeat to the standard cycle before accepting a sketch.
Confusing the graph’s horizontal period with its vertical range.
What are the maximum and minimum values of either basic function?
Worked Examples
A graph can be reconstructed from five anchor points in one cycle.
Plot midline, maximum, midline, minimum, and midline in order.
For a sine cycle, place the five anchors in order and use a smooth curve through them.
Drawing straight line segments that hide the smooth change.
Why are five anchors enough for one basic cycle?
Common Mistakes
Graph reading depends on scale, anchor order, and periodic repetition.
Check the axis measure before reading a zero, peak, or period.
Read the horizontal units, locate anchors, and compare matching stages one cycle apart.
Assuming a full cycle always spans the same x-length after a transformation.
What graph evidence identifies a period?
AP Exam Tips
Use the unit-circle anchor table as a fast sketching tool.
Use the Questionbank set below to test this skill.
Label the start value, first direction, extrema, and period.
Explain one graph feature with its unit-circle coordinate instead of relying on visual appearance alone.
Describing a graph as “looks right” with no anchor evidence.
Can you name the graph’s first two anchor points?
Can you transfer the method?
Describe the direction and concavity of for . Use output change and rate change as separate evidence.
Compare your reasoning
Cosine decreases from toward . Its slopes become more negative as the angle approaches , so its rate decreases and the graph is concave down. The positive output values do not determine concavity.
Work independently, then compare the mathematical evidence in your answer.