Unit 3: Trigonometric and Polar Functions

AP Precalculus: 51 practice questions with detailed explanations.

Unit Study Guide

Core idea

Trigonometry measures rotation: the unit circle turns angles into coordinates, periodic functions turn those coordinates into waves, and polar coordinates replace (x, y) with distance-and-angle.

The unit circle

At angle t, the point on the unit circle is (cos t, sin t); tan t = sin/cos. Know exact values at multiples of π/6 and π/4, extending signs by quadrant. One revolution = 2pi radians = 360 degrees; convert by multiplying degrees by π/180. The identity sin² + cos² = 1 unlocks equation solving.

Sinusoidal graphs

y = A sin(B(x - C)) + D: |A| sets amplitude (height above midline), period = 2pi/B, C shifts phase, D lifts the midline. Model seasons, tides, and ferris wheels by reading these four knobs from context.

Inverse trig

arcsin returns THE angle whose sine matches, restricted to [-π/2, π/2] (arccosine uses [0, π]); other solutions follow from periodicity.

Polar coordinates

(r, θ) converts to rectangular by x = r cos(θ), y = r sin(θ); back-conversion uses r² = x² + y² and tan(θ) = y/x with quadrant care. Families worth recognizing: circles (r = a), cardioids (a ± a cos or sin), roses (r = a cos(n*θ)).

Top 5 Concepts to Master

  1. 1Evaluate trig functions at special angles in any quadrant using reference angles.
  2. 2Graph sinusoids identifying the effects of A, B, C, and D.
  3. 3Solve trig equations using the Pythagorean identity and periodicity.
  4. 4Convert between rectangular and polar coordinates correctly across quadrants.
  5. 5Recognize circle, cardioid, and rose polar graphs from their equations.

Key Terms & Definitions

Practice with Flashcards
Radian

Angle subtending arc length equal to radius; π radians = 180 degrees.

Reference angle

Acute angle to the nearest x-axis, used to extend values to all quadrants.

Periodic function

Repeats outputs at fixed input intervals; trig period = 2pi/B.

Amplitude

|A|, the distance from midline to peak of a sinusoid.

Pythagorean identity

sin²(t) + cos²(t) = 1, valid for every angle.

Inverse trig function

Recovers an angle from a ratio, defined on a restricted domain for uniqueness.

Polar coordinates

(r, θ) locates a point by signed distance and angle from the pole.

Rose curve

r = a cos(n θ) family; petal count depends on parity of n.

Common Misconceptions: Exam Traps

arcsin x equals 1/sin x.

Correct: arcsin denotes the INVERSE FUNCTION (angle recovery); the reciprocal 1/sin x is cosecant.

Amplitude equals B.

Correct: Amplitude is |A|; B controls PERIOD through 2pi/B.

Calculator angle mode does not matter.

Correct: Radian-degree mismatch is the most common trig error; match mode to the problem.

arctan(y/x) alone determines the quadrant.

Correct: Arctangent reads only the line direction; check signs of x and y to place the angle.

Negative r is impossible in polar coordinates.

Correct: (-r, θ) plots the reflection through the pole - equivalent to (r, θ + π).

Featured Practice Questions

Question Bank Breakdown

By difficulty

easy 29medium 22

By topic

Tangent and Other Trigonometric Functions 9Sinusoidal Function Transformations 8Sine and Cosine Function Values 8Sine and Cosine Function Graphs and Properties 7Inverse Trigonometric Functions 6Trigonometric Equations and Identities 6Polar Functions 5Periodic Phenomena 2

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