AP Precalculus Unit 3 Notes

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

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

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

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

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

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

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.

The ray on the x-axis.

An angle's other ray in standard position.

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

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

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.

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

An interval where the polar function is increasing.

An interval where the polar function is decreasing.

The maximum and minimum values of a polar function.

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

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

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

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

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

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