AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
Unit 3B — Review
Notes — Concept Synthesis Across Topics 3.8 – 3.15
💡 Review Goals
By the end of this review you will be able to:
- Connect every skill in Unit 3B into a unified framework
- Move fluidly between trig, inverse trig, and polar coordinate systems
- Identify which topic a question is testing from its surface features
- Avoid the most common errors that AP graders flag in this unit
1. The Unit at a Glance
Topic | Big Idea |
|---|---|
3.8 | Tangent: ratio sin/cos, period π, asymptotes at π/2 + kπ. |
3.9 | Inverse trig functions on restricted ranges. |
3.10 | Solve trig equations on intervals; +2πk or +πk for general. |
3.11 | csc, sec, cot are reciprocals of sin, cos, tan. |
3.12A | Pythagorean, reciprocal, quotient identities. |
3.12B | Sum, difference, double-angle identities. |
3.13 | Polar coordinates (r, θ); convert with x = r cos θ, y = r sin θ. |
3.14A | Polar graphs: circles, limaçons, cardioids, roses. |
3.14B | Pole crossings, max/min r, intersections of polar curves. |
3.15 | Average rate of change of r = f(θ); increasing/decreasing intervals. |
2. Identities Cheat Sheet
Pythagorean identities:
- sin²θ + cos²θ = 1
- tan²θ + 1 = sec²θ
- 1 + cot²θ = csc²θ
Reciprocal/quotient:
- csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ = cos θ/sin θ
Sum/Difference:
- sin(α ± β) = sin α cos β ± cos α sin β
- cos(α ± β) = cos α cos β ∓ sin α sin β (note the sign flip)
Double-angle:
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1
3. Inverse Trig Range Cheat Sheet
Function | Domain | Range |
|---|---|---|
arcsin x | [−1, 1] | [−π/2, π/2] |
arccos x | [−1, 1] | [0, π] |
arctan x | all reals | (−π/2, π/2) |
4. Polar Quick Reference
- Polar to rectangular: x = r cos θ, y = r sin θ
- Rectangular to polar: r² = x² + y², tan θ = y/x (mind the quadrant)
- Circles: r = a (origin), r = a cos θ (offset along x), r = a sin θ (offset along y)
- Limaçon r = a + b cos θ: cardioid when |a/b| = 1; inner loop when |a/b| < 1
- Rose r = a cos(nθ): n petals if n odd, 2n petals if n even
5. Top AP-Style Pitfalls
⚠️ Common mistakes
- Using the wrong period: tangent's period is π, not 2π.
- Treating arcsin as 1/sin or arctan as 1/tan.
- Forgetting the inverse-trig restrictions when composing — arcsin(sin(5π/6)) is π/6, not 5π/6.
- Dividing by sin(x) or cos(x) when solving an equation, losing solutions.
- In equations with sin(bx), forgetting that solutions repeat b times more often than expected.
- Forgetting to check the pole separately when finding intersections of polar curves.
- Plugging negative r as if it were positive when converting to rectangular.
6. Decision Tree for Trig Equations
- LINEAR in one trig function (sin, cos, tan, etc.) → use reference angles, then +2πk or +πk for general
- MULTIPLE-ANGLE (sin(bx), cos(bx)) → solve for the inner argument first, then divide; expect b times more solutions
- QUADRATIC IN A TRIG → substitute u, factor, back-substitute
- INVOLVES sin AND cos OF THE SAME ANGLE → try a Pythagorean identity to eliminate one of them
- MIXED FUNCTIONS → rewrite in terms of sin and cos to begin
- HAS sin(2x) OR cos(2x) → consider a double-angle identity to expand
7. Worked Mini-Example
📘 Example — Multi-step problem
Solve sin(2x) = cos(x) on [0, 2π).
- Use double-angle: 2 sin x cos x = cos x
- Move to one side: 2 sin x cos x − cos x = 0
- Factor: cos x · (2 sin x − 1) = 0
- Either cos x = 0 ⇒ x = π/2, 3π/2 OR sin x = 1/2 ⇒ x = π/6, 5π/6
- Final solutions: {π/6, π/2, 5π/6, 3π/2}
8. Final Reminders
- Memorize identities — you will lose points for incorrect ones
- State the interval at which you are solving every trig equation
- For polar, always work with (r, θ) order; check pole crossings separately
- Average rate of change is just slope — but in polar, it has a geometric interpretation
- Sketch first when unsure; the picture often makes the algebra obvious