AP PRECALCULUS — UNIT 3A · TRIGONOMETRIC & POLAR FUNCTIONS
3.3A — Sine and Cosine Function Values
Notes — The Unit Circle as a Value Table
💡 Learning Objectives (3.3.A Part 1)
By the end of this lesson you will be able to:
- Determine exact values of sin(θ) and cos(θ) for common angles
- Use coterminal and reference angles to evaluate sine and cosine outside [0, 2π]
- Apply the even/odd symmetry of cosine and sine
- Recognize sine and cosine as functions with domain all real numbers
1. A Rotating Point View
Imagine a point P starting at (1, 0) on the unit circle and moving counter-clockwise as the input θ grows. At every θ, P has coordinates (cos θ, sin θ). That single picture contains every value you will ever need for sine and cosine.
Because the circle is closed, after one full revolution (2π radians) the point returns to where it started. So sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ) for every real θ. Both functions are PERIODIC with period 2π.
2. Exact Values on the Unit Circle
Here are the exact (cos, sin) coordinates at every common angle in one rotation. This table is worth knowing fluently.
θ | 0 | π/6 | π/4 | π/3 | π/2 |
|---|---|---|---|---|---|
cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
θ | 2π/3 | 3π/4 | 5π/6 | π |
|---|---|---|---|---|
cos θ | −1/2 | −√2/2 | −√3/2 | −1 |
sin θ | √3/2 | √2/2 | 1/2 | 0 |
θ | 7π/6 | 5π/4 | 4π/3 | 3π/2 |
|---|---|---|---|---|
cos θ | −√3/2 | −√2/2 | −1/2 | 0 |
sin θ | −1/2 | −√2/2 | −√3/2 | −1 |
θ | 5π/3 | 7π/4 | 11π/6 | 2π |
|---|---|---|---|---|
cos θ | 1/2 | √2/2 | √3/2 | 1 |
sin θ | −√3/2 | −√2/2 | −1/2 | 0 |
3. Using Reference Angles
For any angle outside [0, 2π], first reduce it to [0, 2π) using coterminality, then apply the quadrant-sign rule. Reference angles let you reuse the Quadrant-I table above.
📘 Example — sin(13π/6) and cos(−π/3)
- 13π/6 − 2π = π/6 (coterminal). But 13π/6 is in Quadrant IV, where sine is negative. Reference angle = π/6. sin(π/6) = 1/2. So sin(13π/6) = −1/2
- −π/3: Quadrant IV (cos positive). Reference angle = π/3. cos(π/3) = 1/2. So cos(−π/3) = 1/2
4. Even and Odd Symmetries
Notice that (cos(−θ), sin(−θ)) is the reflection of (cos θ, sin θ) across the x-axis. Reflecting flips the y-coordinate but leaves the x-coordinate alone. That gives us:
- cos(−θ) = cos(θ) (cosine is EVEN — symmetric across the y-axis)
- sin(−θ) = −sin(θ) (sine is ODD — symmetric about the origin)
📘 Example — Applying symmetry
- sin(−π/4) = −sin(π/4) = −√2/2
- cos(−5π/6) = cos(5π/6) = −√3/2
5. Bounds on Sine and Cosine
Because (cos θ, sin θ) is always a point on the UNIT circle, both coordinates are bounded:
−1 ≤ cos(θ) ≤ 1 and −1 ≤ sin(θ) ≤ 1 for every real θ
This is a very useful fact for bounding expressions, solving inequalities, and checking your work.
6. Domain and Periodicity
- Domain of sine: all real numbers
- Domain of cosine: all real numbers
- Period of sine and cosine: 2π
- Range of both: [−1, 1]
7. Summary
- (cos θ, sin θ) is the point on the unit circle at angle θ from the positive x-axis
- Both functions are periodic with period 2π; both output values in [−1, 1]
- Cosine is even: cos(−θ) = cos(θ); Sine is odd: sin(−θ) = −sin(θ)
- Use coterminality (add or subtract 2π) and reference angles to evaluate at any input