AP Exam Prep · AP Precalculus

AP Precalculus Unit 3 Practice Test: Trigonometric and Polar Functions

Unit 3 models repeating patterns. It starts with the unit circle and the sine, cosine and tangent functions, then builds sinusoidal models from amplitude, period, phase shift and midline. It finishes with inverse trig, equations and identities, the reciprocal functions, and polar coordinates and polar graphs.

Exam weighting:: 30–35%
Questions:: 22 multiple choice
Time:: about 55 minutes
Cost:: Free, no sign-up

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Question 1 of 22 · 0 answered

Topic 3.1 · Apply it

The height of a seat on a turning Ferris wheel is periodic with a period of 40 seconds. At t = 5 seconds the seat is 30 metres above the ground. How high is it at t = 85 seconds?

60 metresIt cannot be determined without the radius.510 metres30 metres

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Topic 3.7 · Data analysis

Average monthly temperature in a hypothetical city
Month m	1 (Jan)	4 (Apr)	7 (Jul)	10 (Oct)
Temperature (°C)	10	20	30	20

The temperature follows a sinusoidal pattern with a period of 12 months, lowest in January and highest in July. Which model fits?

T(m) = −10cos((π/12)(m − 1)) + 20T(m) = −20cos((π/6)(m − 1)) + 10T(m) = −10cos((π/6)(m − 1)) + 20T(m) = 10cos((π/6)(m − 1)) + 20

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What AP Precalc Unit 3 covers

Unit 3, Trigonometric and Polar Functions, makes up 30–35% of the multiple-choice section and usually takes ~35–50 class periods. These are the topics in the current course framework, and every question in the test above is tagged with one of them.

3.1Periodic Phenomena
3.2Sine, Cosine, and Tangent
3.3Sine and Cosine Function Values
3.4Sine and Cosine Function Graphs
3.5Sinusoidal Functions
3.6Sinusoidal Function Transformations
3.7Sinusoidal Function Context and Data Modeling
3.8The Tangent Function
3.9Inverse Trigonometric Functions
3.10Trigonometric Equations and Inequalities
3.11The Secant, Cosecant, and Cotangent Functions
3.12Equivalent Representations of Trigonometric Functions
3.13Trigonometry and Polar Coordinates
3.14Polar Function Graphs
3.15Rates of Change in Polar Functions

Unit 3 key ideas

The unit circle defines sine and cosine

For an angle θ in standard position, the point where its terminal ray meets the unit circle is (cos θ, sin θ), and tan θ = sin θ ÷ cos θ is the slope of the ray. Learn the values at multiples of π/6 and π/4.

Read a sinusoid in four numbers

In f(θ) = a·sin(b(θ + c)) + d: |a| is the amplitude, 2π ÷ |b| is the period, d is the midline and −c is the phase shift. The amplitude is half the distance from maximum to minimum, and the midline is halfway between them.

Inverse trig has restricted outputs

arcsin and arctan return angles in [−π/2, π/2], and arccos returns angles in [0, π]. To solve sin θ = k on a longer interval, find the principal value and then use symmetry and the period to find every solution.

Polar coordinates measure angle and signed distance

A point (r, θ) is r units from the origin along the ray at angle θ (opposite direction if r < 0), so x = r cos θ and y = r sin θ. In r = f(θ), the distance from the origin increases when r is positive and increasing or negative and decreasing.

Common Unit 3 mistakes

Using a period of b instead of 2π ÷ b for sin(bθ).
Giving only the calculator's principal value when an equation asks for every solution on an interval.
Mixing degrees and radians; the course uses radians unless degrees are stated.

How to study this unit

Sketch one full cycle and mark the maximum, minimum and midline crossings before answering a graph question.
For polar questions, convert a few points to (x, y) to see where the graph is.

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More AP Precalculus practice tests

← Unit 2: Exponential and Logarithmic Functions Unit 4: Functions Involving Parameters, Vectors, and Matrices →

About these questions

StudyCoach wrote these questions for this test. Each one is mapped to a topic in the College Board's current AP Precalculus course framework. Every answer key was also checked by a separate AI review, and any disagreement was rechecked by hand. They have not yet been reviewed by an AP Precalculus teacher. If you think an answer is wrong, email support@studycoach.io.

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