AP PRECALCULUS — UNIT 3A · TRIGONOMETRIC & POLAR FUNCTIONS
Unit 3A — Review
Notes — Concept Synthesis Across Topics 3.1 – 3.7
💡 Review Goals
By the end of this review you will be able to:
- Connect every skill in Unit 3A into a unified framework for trigonometric functions
- Move fluidly between the unit-circle, right-triangle, and graphical views of sine and cosine
- Identify which topic a question is testing from its wording
- Avoid the most common trigonometry errors that come up on the AP exam
1. The Unit at a Glance
Unit 3A moves from a casual look at periodic phenomena to fully-parameterized sinusoidal models. The throughline: the same rotating point on a unit circle secretly powers every formula in the unit.
Topic | Big Idea |
|---|---|
3.1 | Recognize periodic behavior; extract period, midline, amplitude, max/min. |
3.2A | Radians: s = rθ; 180° = π; memorize the unit circle. |
3.2B | Definitions of sin, cos, tan on the unit circle and in right triangles; quadrant signs. |
3.3A | Exact values at common angles; period 2π; even/odd symmetry. |
3.3B | Use Pythagorean identity and reference angles to find values in any quadrant. |
3.4 | Graphs of y = sin(x), y = cos(x); five key points, concavity, symmetry. |
3.5 | General sinusoidal form y = a sin(b(x − c)) + d; amplitude, period, midline, phase. |
3.6A | Vertical transformations: amplitude scaling, vertical shifts, reflection. |
3.6B | Horizontal transformations: period change, phase shift; factor b before reading c. |
3.7 | Sinusoidal models in context: map max/min/period/reference to parameters. |
2. The Most Important Identities
- cos²(θ) + sin²(θ) = 1 (Pythagorean)
- sin(−θ) = −sin(θ) and cos(−θ) = cos(θ) (odd / even)
- sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ) (period 2π)
- cos(x) = sin(x + π/2) (phase shift between cos and sin)
- sin(π − θ) = sin(θ), cos(π − θ) = −cos(θ) (supplement)
- sin(π/2 − θ) = cos(θ), cos(π/2 − θ) = sin(θ) (complement)
3. Exact-Value Cheat Sheet
θ | 0 | π/6 | π/4 | π/3 | π/2 |
|---|---|---|---|---|---|
sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
tan θ | 0 | √3/3 | 1 | √3 | undef |
Extend by quadrant using the sign rule (‘All Students Take Calculus’) and reference angles.
4. Decision Tree for Evaluation
- IS THE ANGLE IN [0, π/2]? → Use the cheat sheet directly.
- IS IT OUTSIDE [0, 2π]? → Subtract 2π until you land in [0, 2π).
- IS IT IN QUADRANT II, III, OR IV? → Find reference angle, then apply sign rule.
- ARE YOU GIVEN ONE TRIG VALUE AND A QUADRANT? → Use Pythagorean identity to get the other.
- ARE YOU GIVEN A POINT (x, y) ON A CIRCLE OF RADIUS r? → cos = x/r, sin = y/r.
5. Reading a Sinusoidal Graph
When you see a sinusoidal graph on the exam, extract parameters in this order:
- Max and min values → d = (max + min)/2 and |a| = (max − min)/2
- One full period length P → b = 2π/P
- Pick a reference point that is easy to spot. Max ⇒ cosine with a > 0. Min ⇒ cosine with a < 0. Midline crossing going up ⇒ sine with a > 0.
- That reference point’s x-coordinate is c.
- Write the model y = a · (sin or cos)(b(x − c)) + d and verify at one more point.
6. Top Pitfalls
⚠️ Common mistakes
- Using degrees where radians are required (especially on calculus-style problems and the formula s = rθ).
- Mis-reading amplitude as the full max-to-min swing (it is HALF that).
- Measuring period from a peak to the next TROUGH instead of peak-to-peak (that's half a period).
- Forgetting to factor b before identifying c (e.g. treating sin(2x − π) as a shift of π instead of π/2).
- Ignoring quadrant signs when using the Pythagorean identity.
7. A Worked Review Example
📘 Example — Tying it all together
A wave has maxima of y = 11 every 6 seconds. At t = 1 the graph is on its midline and rising. Write a sinusoidal model and find the minimum value.
- Max is 11. ‘Every 6 seconds’ = period 6.
- ‘Midline rising at t = 1’ suggests sine with c = 1.
- We still need the min. With midline crossing rising at t = 1, the max occurs a quarter-period later: t = 1 + 6/4 = 2.5. So max = 11 at t = 2.5.
- From the max/min pattern, if we assume the midline is halfway, we need another data point. Assuming amplitude 4 gives min = 3 and midline = 7, so y = 4 sin((π/3)(t − 1)) + 7.
- Minimum value of the model is 7 − 4 = 3.
8. Final Reminders
- Always work in radians unless the problem explicitly says degrees
- When writing sinusoidal models, put them in the form a · (sin or cos)(b(x − c)) + d
- Check your model against at least two known values before using it for prediction
- Know your exact values cold — you'll need them on the no-calculator section
- Don't forget the period in general solutions: +2πk for sine/cosine equations