AP PRECALCULUS — UNIT 3A · TRIGONOMETRIC & POLAR FUNCTIONS
3.4 — Sine and Cosine Function Graphs
Notes — From Unit Circle to Wave
💡 Learning Objectives (3.4.A)
By the end of this lesson you will be able to:
- Sketch y = sin(x) and y = cos(x) over several periods
- Identify the five key points (starting point, quarter points, half-period, three-quarter, full period)
- Describe where each function is increasing, decreasing, and changing concavity
- Relate graph features back to the rotating-point picture on the unit circle
1. Building y = sin(x) from the Unit Circle
As θ rotates from 0 to 2π, the y-coordinate of the point (cos θ, sin θ) starts at 0, rises to 1 at π/2, falls back to 0 at π, continues to −1 at 3π/2, and returns to 0 at 2π. Plot θ on the horizontal axis and that y-value on the vertical axis — the result is the SINE CURVE.
Because θ is periodic with period 2π, the same shape repeats indefinitely to the left and right. The range is [−1, 1] and the domain is all reals.
2. Building y = cos(x)
Now track the x-coordinate of the same rotating point. It starts at 1, falls to 0 at π/2, continues down to −1 at π, rises back to 0 at 3π/2, and returns to 1 at 2π. The COSINE CURVE is the sine curve shifted π/2 units to the left — they have the same shape.
3. Five Key Points on One Period
Over the interval [0, 2π], each curve is pinned by five evenly-spaced ‘key points’:
x | 0 | π/2 | π | 3π/2 | 2π |
|---|---|---|---|---|---|
sin x | 0 | 1 | 0 | −1 | 0 |
cos x | 1 | 0 | −1 | 0 | 1 |
Knowing these five (x, y) pairs is enough to sketch either curve accurately. All other points lie on a smooth wave connecting them.
4. Increasing / Decreasing Behavior
For y = sin(x) on [0, 2π]:
- Increasing on (0, π/2) — sine climbs from 0 to 1
- Decreasing on (π/2, 3π/2) — sine falls from 1 down through 0 to −1
- Increasing on (3π/2, 2π) — sine rises from −1 back to 0
For y = cos(x) on [0, 2π]:
- Decreasing on (0, π) — cosine falls from 1 to −1
- Increasing on (π, 2π) — cosine rises from −1 back to 1
5. Concavity Behavior
Each curve has two concave-up arcs and two concave-down arcs per period. Look for places where the curve is at a maximum, minimum, or zero-crossing — these are where concavity flips.
For y = sin(x) on [0, 2π]:
- Concave down on (0, π) — curve is above the x-axis, bending like a frown
- Concave up on (π, 2π) — curve is below the x-axis, bending like a smile
📝 Inflection points
Every zero of y = sin(x) is also an INFLECTION POINT — the place where the concavity flips. Same is true for y = cos(x).
6. Symmetry of the Graphs
- y = sin(x) has ODD symmetry: the graph is symmetric about the origin. This matches sin(−x) = −sin(x).
- y = cos(x) has EVEN symmetry: the graph is symmetric about the y-axis. This matches cos(−x) = cos(x).
7. Horizontal Shift Between Sine and Cosine
The cosine graph is the sine graph shifted LEFT by π/2. That is:
cos(x) = sin(x + π/2) and sin(x) = cos(x − π/2)
This is why they are called ‘phase-shifted’ copies of each other — they carry the same wave with different starting points.
8. Numerical Sanity Check
Use these quick values to remember the graph shape at a glance:
x | −π | −π/2 | 0 | π/2 | π | 3π/2 | 2π |
|---|---|---|---|---|---|---|---|
sin x | 0 | −1 | 0 | 1 | 0 | −1 | 0 |
cos x | −1 | 0 | 1 | 0 | −1 | 0 | 1 |
9. Summary
- y = sin(x) and y = cos(x) are periodic with period 2π and range [−1, 1]
- Each period has five key points evenly spaced by π/2
- Sine is odd (origin symmetry); cosine is even (y-axis symmetry)
- Cosine = sine shifted left by π/2
- Zeros of each function are also inflection points