Comprehensive Guide to Trigonometric & Polar Functions

Periodic Phenomena

Events or processes that repeat in identical intervals of time or space.

Period

The smallest distinct positive value of p such that f(x + p) = f(x) for all x.

Cycle

One complete repetition of the pattern in a periodic function.

Midline

The horizontal line that runs in the middle of the graph's maximum and minimum values.

Amplitude

The vertical distance from the midline to a maximum or minimum; indicates strength or height of the wave.

Concave Up

When the graph bends upward, indicating that the rate of change is increasing.

Concave Down

When the graph bends downward, indicating that the rate of change is decreasing.

Standard Position

An angle is in standard position when its vertex is at the origin and initial side lies along the positive x-axis.

Coterminal Angles

Angles that end at the same terminal side but differ by full rotations.

Radians

An angle measure that relates directly to arc length; one radian is the angle subtended by an arc equal to the radius of the circle.

Unit Circle

A circle with radius r=1 centered at the origin, where cos(θ) = x and sin(θ) = y.

30-60-90 Triangle

A special triangle with side ratios of 1:√3:2; used for finding trigonometric values.

45-45-90 Triangle

A special triangle with side ratios of 1:1:√2; used for finding trigonometric values.

Reference Angle

The acute angle formed by the terminal side of an angle and the x-axis.

Sine Graph

A periodic graph starting at the midline (0,0) going up, with a domain of (-∞, ∞) and a range of [-1, 1].

Cosine Graph

A periodic graph starting at an extremum (0,1) going down, with a domain of (-∞, ∞) and a range of [-1, 1].

Frequency

The number of cycles per unit interval of input, related to the period.

Transformations

Changes to sine and cosine functions represented by the parameters a, b, c, and d in the general sine and cosine forms.

Midline (y=d)

The line representing the vertical center of a sine or cosine wave.

Tangent Function

The function defined as f(x) = tan(x) = sin(x)/cos(x) with a period of π.

Inverse Sine Function

The function y = arcsin x with a restricted domain of [-π/2, π/2].

Inverse Cosine Function

The function y = arccos x with a restricted domain of [0, π].

Inverse Tangent Function

The function y = arctan x with a restricted domain of (-π/2, π/2).

Pythagorean Identities

Key identities including sin²θ + cos²θ = 1.

Sum and Difference Formula

Equations used to find exact values of angles, such as sin(A ± B) = sin A cos B ± cos A sin B.

Polar Coordinates

A system defining a point by directed distance r from the origin and angle θ from the positive x-axis.

Conversion to Rectangular

Formulas to convert from polar to rectangular coordinates: x = r cos(θ), y = r sin(θ).

Limaçons

Polar graphs defined as r = a ± b cos(θ) or r = a ± b sin(θ), characterized by shapes influenced by the ratio of a to b.

Rose Curves

Polar graphs of the form r = a cos(nθ) or r = a sin(nθ), with the number of petals depending on n.

Rate of Change in Polar Functions

The average rate of change calculated as ROC = Δr/Δθ.

Relative Maximum

When the rate of change transitions from positive to negative in polar functions, indicating the maximum distance from the pole.

Trigonometric Equations

Equations involving trigonometric functions that can be solved by isolating the trig function and finding reference angles.

Cosecant Function

Reciprocal of sine, defined as csc(θ) = 1/sin(θ) with asymptotes at x = nπ.

Secant Function

Reciprocal of cosine, defined as sec(θ) = 1/cos(θ) with asymptotes at x = (π/2) + nπ.

Cotangent Function

Reciprocal of tangent, defined as cot(θ) = cos(θ)/sin(θ) with asymptotes at x = nπ.

Units in Trigonometry

Most AP Calculus questions assume radians unless explicitly stated otherwise.

Key Features of Sine Functions

The sine function is an odd function symmetric about the origin.

Key Features of Cosine Functions

The cosine function is an even function symmetric about the y-axis.

Graphing Sinusoidal Functions

Steps include drawing the midline, determining max/min, and identifying key points.

Asymptotes of Tangent Function

Vertical asymptotes occur where cos(x) = 0.

Graph Features of Tangent Function

The tangent function has a period of π and no amplitude.

General Solutions for Sin and Cos

Solutions incorporate the period of the function; sin and cos include +2πn, while tangent includes +πn.

Transformation Effects on Sine and Cosine

Vertical stretch/compression, horizontal stretch/compression, and shifts caused by parameters a, b, c, and d.

Analyzing Frequency in Sinusoidal Graphs

The reciprocal relationship between frequency and period in trigonometric functions.

Rate of Change Interpretation

Understanding increases and decreases of radial distance in terms of angle rotation.

Common Mistakes in Trigonometry

Errors such as failing to factor or confusing units when working with angles.