AP Precalculus Unit 3 Notes

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Periodic Phenomena

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50 Terms

1

Periodic Phenomena

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

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2

Periodic Function

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

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3

Period

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

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4

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.

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5

Concavity

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

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6

Average Rate of Change

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

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7

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.

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8

Initial side

The ray on the x-axis.

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Terminal side

An angle's other ray in standard position.

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10

Positive Angle

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

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Negative Angle

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

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12

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.

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Coterminal Angles

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

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14

Special Triangles

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

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15

Quadrant Positivity

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

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Phase Shift

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

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Sinusoidal Functions

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

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Sign Determination

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

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19

Graph of Sine Function

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

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Graph of Cosine Function

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

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21

Transformations of Sine and Cosine Functions

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

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22

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.

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23

Interpreting, Verifying, and Reporting with Models

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

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Tangent Function

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

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25

Additive and Multiplicative Transformations

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

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Tangent Function

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

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Period of Tangent Function

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

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Transformations of Tangent Function

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

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Key Features of Tangent Function

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

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30

Inverse Trigonometric Functions

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

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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.

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Solving Trigonometric Equations

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

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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.

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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.

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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.

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36

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.

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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.

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38

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.

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Average rate of change

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

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Δr

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

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Δθ

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

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Increasing interval

An interval where the polar function is increasing.

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Decreasing interval

An interval where the polar function is decreasing.

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Extrema

The maximum and minimum values of a polar function.

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Distance from origin

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

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r positive

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

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r negative

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

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r increasing

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

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r decreasing

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

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50

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.

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