Mastering Trigonometric Functions in AP Precalculus

Periodic Function

A function is periodic if there exists a positive number p such that f(x + p) = f(x) for all x.

Fundamental Period

The smallest positive value p for which a periodic function repeats.

Cycle

One complete repetition of the pattern of a periodic function.

Unit Circle

A circle centered at the origin with a radius of 1, used to define trigonometric functions.

Angle in Standard Position

An angle measured from the positive x-axis, rotating counter-clockwise for positive angles.

Radians

A unit of angular measure where one full revolution is equal to 2π radians.

Arc Length

On the unit circle, the radian measure of an angle is equal to the length of the arc intercepted by that angle.

Cosine Function

The x-coordinate of a point on the unit circle corresponding to an angle θ.

Sine Function

The y-coordinate of a point on the unit circle corresponding to an angle θ.

Tangent Function

The ratio of the y-coordinate to the x-coordinate, tan(θ) = sin(θ)/cos(θ), where x ≠ 0.

ASTC Mnemonic

A mnemonic to remember in which quadrants sine, cosine, and tangent are positive.

Pythagorean Identity

The identity stating that cos²(θ) + sin²(θ) = 1.

Sine Graph

The graph of the function y = sin(x), which oscillates between -1 and 1.

Cosine Graph

The graph of the function y = cos(x), starting at a maximum of 1 when x = 0.

Amplitude

The maximum distance from the midline to the peak or trough of a sinusoidal graph.

Period of a Function

The length of one complete cycle in a periodic function, calculated as P = 2π/|b|.

Vertical Shift

The displacement of a sinusoidal graph up or down, represented by parameter d.

Phase Shift

The horizontal shift of a sinusoidal graph determined by the value of c in the equation.

Concave Up

Intervals of the graph where it opens upwards; for sine, it occurs on (π, 2π).

Concave Down

Intervals of the graph where it opens downwards; for sine, it occurs on (0, π).

Inflection Points

Points on the graph where the concavity changes.

Modeling Data with Sinusoidal Functions

Using the equation y = a sin(b(x - c)) + d to represent periodic phenomena.

Frequency

The number of complete cycles that occur in a unit of time, related to the parameter b.

Maximum Height in a Ferris Wheel Model

The centered height of the Ferris wheel plus the amplitude, calculated as d + a.

Minimum Height in a Ferris Wheel Model

The centered height of the Ferris wheel minus the amplitude, calculated as d - a.

Common Mistake #1

Using degrees instead of radians; AP Precalculus relies primarily on radian measure.

Common Mistake #2

Incorrectly determining phase shift without factoring out b; it must be done to find the true shift.