AP Precalculus, AP exams, 30–35%

Unit 3: Trigonometric and Polar Functions

Unit 3 covers periodic phenomena through the trigonometric functions and extends into polar coordinates. Sinusoidal modeling, inverse trigonometric functions, identities, and equations are the core, and polar graphs bring in rates of change of the radius with respect to angle. This unit is the most notation-heavy of the three assessed units.

Position: 3 of 4 units
Published weighting: 30–35% in the official framework
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Precalculus. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Periodic phenomena and the unit circle: Radian measure, the unit circle, and defining sine, cosine, and tangent from it.
Sinusoidal functions and transformations: Amplitude, period, phase shift, and midline, and constructing a sinusoid from described behavior.
Sinusoidal modeling: Fitting a sinusoidal model to periodic data and interpreting its parameters in context.
Tangent, reciprocal, and inverse functions: Graphs of tangent, secant, cosecant, and cotangent, and the restricted domains inverse functions require.
Trigonometric identities and equations: Pythagorean, sum, difference, and double-angle identities, and solving equations over a given interval.
Polar functions: Converting between polar and rectangular coordinates, graphing polar curves, and describing how the radius changes with angle.

Study tips

How to practice Trigonometric and Polar.

Section-specific guidance that follows from how this material is usually tested.

Rebuild the unit circle from memory regularly rather than relying on a calculator; Part A of each section has none.
Practice extracting all four sinusoidal parameters from a described context, which is a standard free-response setup.
Describe polar graphs in words, naming where the radius increases, decreases, or is negative.

Where this fits

Unit 3: Trigonometric and Polar Functions is 3 of 4 units in the AP Precalculus guide, and it carries a published weighting of 30–35% in the official framework. Use the full guide to balance it against everything else you have to review.

Back to the AP Precalculus guide Official exam information

Sample Trigonometric and Polar questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
What is the exact value of sin(π/6)?
1. A1/2
2. B√2/2
3. C√3/2
4. D1
Show answer
Answer: A. 1/2
The angle π/6 radians is 30 degrees, and on the unit circle its terminal point is (√3/2, 1/2). The sine is the y-coordinate, so sin(π/6) = 1/2. The value √3/2 is cos(π/6), the x-coordinate, which is the usual mix-up.
Key insight: On the unit circle the sine is the y-coordinate and the cosine is the x-coordinate.
Question 2Foundational
What is the period of the function f(x) = 3sin(2x) + 1?
1. Aπ
2. B2π
3. Cπ/2
4. D4π
Show answer
Answer: A. π
For f(x) = a sin(bx) + d the period is 2π/|b|, and here b = 2, so the period is π. The amplitude 3 and the vertical shift 1 stretch and lift the graph but never change how long one cycle takes.
Key insight: Only the coefficient of x affects the period; amplitude and vertical shift do not.
Question 3Intermediate
A sinusoidal function models the height of a rider on a Ferris wheel whose lowest point is 5 meters above the ground and whose highest point is 45 meters above the ground. What is the amplitude of the model?
1. A20 meters
2. B25 meters
3. C40 meters
4. D45 meters
Show answer
Answer: A. 20 meters
The amplitude is half the distance between the maximum and minimum: (45 - 5)/2 = 20 meters. The value 25 is the midline, the average of the two heights, and 40 is the full vertical range rather than half of it.
Key insight: Amplitude is half the peak-to-trough distance; the midline is their average.
Question 4Intermediate
What is the rectangular equation of the polar equation r = 4?
1. Ax^2 + y^2 = 16
2. Bx^2 + y^2 = 4
3. Cx + y = 4
4. Dy = 4
Show answer
Answer: A. x^2 + y^2 = 16
Because r^2 = x^2 + y^2, the equation r = 4 becomes x^2 + y^2 = 16, the circle of radius 4 centered at the origin. Writing x^2 + y^2 = 4 forgets to square the radius, describing a circle of radius 2 instead.
Key insight: Convert with r^2 = x^2 + y^2, which means a constant r must be squared.
Question 5Advanced
If cos(θ) = 3/5 and θ lies in the fourth quadrant, what is the value of sin(2θ)?
1. A-24/25
2. B24/25
3. C-7/25
4. D-12/25
Show answer
Answer: A. -24/25
The Pythagorean identity gives sin(θ) = -4/5 because sine is negative in the fourth quadrant. The double-angle identity then gives sin(2θ) = 2sin(θ)cos(θ) = 2(-4/5)(3/5) = -24/25. The value -7/25 is cos(2θ), and dropping the factor of 2 gives -12/25.
Key insight: Fix the sign from the quadrant before applying a double-angle identity.

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Unit 3: Trigonometric and Polar Functions