AP Precalculus Unit 3 Notes

Periodic Phenomena

Occurrences or relationships that display a repetitive pattern over time or space.

Periodic Function

A function that replicates a sequence of y-values at fixed intervals.

Period

The gap between repetitions of a periodic function, representing the length of one complete cycle.

Intervals of Increase and Decrease

Sets of x-values from the lower to the maximum point and from the upper to the minimum point, respectively.

Concavity

Determines whether the function is concave up or down, influencing its behavior.

Average Rate of Change

Calculated as the change in output divided by the change in input.

Standard position

If an angle's initial side is parallel to the positive x-axis and its vertex is at the origin, it is said to be in the standard position.

Initial side

The ray on the x-axis.

Terminal side

An angle's other ray in standard position.

Positive Angle

If you rotate something counterclockwise, it's considered a positive angle.

Negative Angle

If you rotate something clockwise, it's considered a negative angle.

Radian angle measures

The measure of an angle in radians using the formula θ=s/r, where s is the arc length and r is the circle's radius.

Coterminal Angles

Angles that end up in the same position but have different measures due to multiple rotations around a circle.

Special Triangles

Triangles on the unit circle with specific side length ratios used to evaluate trigonometric functions with exact ratios.

Quadrant Positivity

Determining in which quadrants sine and cosine are positive and identifying positive ratios in each quadrant.

Phase Shift

The horizontal shift in a sine or cosine function upon adding an angle.

Sinusoidal Functions

Sine and cosine functions that share a sinusoidal nature and exhibit the same shape and traits.

Sign Determination

Relating point coordinates on the unit circle to trigonometric functions and understanding how quadrant positions affect the signs of cosine and sine.

Graph of Sine Function

Using unit circle angles for x-axis representation and a coordinate range of [-1,1] for the y-axis.

Graph of Cosine Function

Using unit circle angles for x-axis representation and a coordinate range of [-1,1] for the x-axis.

Transformations of Sine and Cosine Functions

Modifying the critical features of sine and cosine functions through amplitude, vertical shift, period, and phase shift.

Sinusoidal Function

A function in the form y=asin[b(x-c)]+d that represents a sine wave pattern and can be transformed through amplitude, frequency, and midline changes.

Interpreting, Verifying, and Reporting with Models

Selecting and verifying suitable models for periodic phenomena problems and reporting findings with relevant information.

Tangent Function

Constructing representations of the tangent function using the unit circle and describing its key characteristics.

Additive and Multiplicative Transformations

Transformations involving the tangent function that involve adding or multiplying angles.

Tangent Function

The tangent function, f(θ)=tanθ, is a periodic function with a domain and range based on the unit circle.

Period of Tangent Function

The tangent function has a period of π and is undefined where cosθ=0, leading to a discontinuous function.

Transformations of Tangent Function

The tangent function, f(θ)=atan[b(θ−c)]+d, can be transformed by adjusting frequency and midline.

Key Features of Tangent Function

The key features of the tangent function include domain, range, x-intercepts, y-intercept, period, amplitude, and midline.

Inverse Trigonometric Functions

The inverse trigonometric functions, denoted as arcsin, arccos, and arctan, introduce a nuanced understanding of function inverses.

Analytical and Graphical Representations of Inverse Trigonometric Functions

Inverse trigonometric functions can be represented analytically and graphically, such as sin(arcsinx)=x and arcsin(sinx)=x.

Solving Trigonometric Equations

Trigonometric equations involve one or more of the six trigonometric functions and can be solved using inverse functions and algebraic manipulations.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions, including cosecant (csc), secant (sec), and cotangent (cot), relate to the fundamental trigonometric functions and can be used to solve equations.

Characteristics of Reciprocal Trigonometric Functions

The cosecant, secant, and cotangent functions have specific characteristics such as domain, range, x-intercepts, y-intercept, period, amplitude, and midline.

Equivalent Representations of Trigonometric Functions

Trigonometric expressions can be rewritten using the Pythagorean identity and sine and cosine sum identities, which can also be used to solve equations.

Trigonometry and Polar Coordinates

Polar coordinates involve measurements of distance (r) from the origin and angle (θ) from the positive horizontal axis, and can be converted from rectangular coordinates.

Graphs of Polar Functions

Polar functions can represent circles, roses, and limacons, each with their own characteristics based on parameters such as radius and petal count.

Rates of Change in Polar Functions

Polar functions have characteristics such as rate of change, intervals of increase and decrease, positive and negative intervals, and extrema.

Average rate of change

The ratio of the change in radius values to the change in θ, indicating how the radius changes per radian.

Δr

The change in radius values between two points on a polar function.

Δθ

The change in θ (angle) between two points on a polar function.

Increasing interval

An interval where the polar function is increasing.

Decreasing interval

An interval where the polar function is decreasing.

Extrema

The maximum and minimum values of a polar function.

Distance from origin

The distance between a point on a polar function and the origin.

r positive

When r is positive, the distance from the origin is increasing.

r negative

When r is negative, the distance from the origin is decreasing.

r increasing

When r is increasing, the distance from the origin is increasing.

r decreasing

When r is decreasing, the distance from the origin is decreasing.

Average rate of change formula

Δr/Δθ = (f(θ2) - f(θ1)) / (θ2 - θ1), calculates the average rate of change of r for θ over an interval.