Studied by 427 people

5.0(1)

get a hint

hint

1

Periodic Phenomena

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

New cards

2

Periodic Function

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

New cards

3

Period

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

New cards

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.

New cards

5

Concavity

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

New cards

6

Average Rate of Change

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

New cards

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.

New cards

8

Initial side

The ray on the x-axis.

New cards

9

Terminal side

An angle's other ray in standard position.

New cards

10

Positive Angle

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

New cards

11

Negative Angle

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

New cards

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.

New cards

13

Coterminal Angles

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

New cards

14

Special Triangles

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

New cards

15

Quadrant Positivity

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

New cards

16

Phase Shift

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

New cards

17

Sinusoidal Functions

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

New cards

18

Sign Determination

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

New cards

19

Graph of Sine Function

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

New cards

20

Graph of Cosine Function

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

New cards

21

Transformations of Sine and Cosine Functions

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

New cards

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.

New cards

23

Interpreting, Verifying, and Reporting with Models

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

New cards

24

Tangent Function

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

New cards

25

Additive and Multiplicative Transformations

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

New cards

26

Tangent Function

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

New cards

27

Period of Tangent Function

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

New cards

28

Transformations of Tangent Function

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

New cards

29

Key Features of Tangent Function

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

New cards

30

Inverse Trigonometric Functions

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

New cards

31

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.

New cards

32

Solving Trigonometric Equations

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

New cards

33

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.

New cards

34

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.

New cards

35

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.

New cards

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.

New cards

37

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.

New cards

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.

New cards

39

Average rate of change

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

New cards

40

Δr

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

New cards

41

Δθ

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

New cards

42

Increasing interval

An interval where the polar function is increasing.

New cards

43

Decreasing interval

An interval where the polar function is decreasing.

New cards

44

Extrema

The maximum and minimum values of a polar function.

New cards

45

Distance from origin

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

New cards

46

r positive

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

New cards

47

r negative

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

New cards

48

r increasing

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

New cards

49

r decreasing

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

New cards

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.

New cards