APPC 3.1-3.5

Periodic Relationship

Repeating pattern over equal-length intervals.

Period

Length of x-values for one complete cycle.

Periodic Function

Function that repeats values over intervals.

Graph of f(x)

Visual representation of periodic function f.

Period of g

Least interval for function g to repeat.

g(x) = g(x + kp)

Function g repeats every k periods.

h(x) Periodicity

Function h repeats every # units.

Horizontal Translation

Shifting function along x-axis without changing shape.

Domain of f

All real numbers for function f.

Output Value

Result of function for a given input.

Input Values for f

Values yielding output of 1 for function f.

Rate of Change

How function value changes over time interval.

Concave Up

Graph shape indicating increasing rate of change.

t1 and t2

Specific time intervals for analyzing h.

h is Positive

Function h remains above zero in interval.

h is Negative

Function h remains below zero in interval.

Increasing Function

Function value rises as input increases.

Decreasing Function

Function value falls as input increases.

Terminal Ray

Second ray of an angle in standard position.

Sine

Ratio of vertical displacement to hypotenuse.

Cosine

Ratio of horizontal displacement to hypotenuse.

Tangent

Ratio of vertical displacement to horizontal displacement.

Point P

Intersection of terminal ray and unit circle.

Coordinates of P

Point P represented as (x, y).

Vertical Displacement

Distance from x-axis to point P.

Horizontal Displacement

Distance from y-axis to point P.

Radian Measure

Angle measure based on arc length over radius.

Sine of Angle

sin(θ) = y/r for point P.

Cosine of Angle

cos(θ) = x/r for point P.

Tangent of Angle

tan(θ) = y/x for point P.

Example Values

Specific sin, cos, tan values for given angles.

Quadrant II Characteristics

sin positive, cos negative, tan negative.

Angle Measure in Standard Position

Measured counterclockwise from positive x-axis.

Positive Angles

Measured counterclockwise from the positive x-axis.

Negative Angles

Measured clockwise from the positive x-axis.

Standard Position

Angle's vertex at origin, initial ray on x-axis.

Periodic Behavior

Angles with same terminal ray differ by revolutions.

Arc Length

Distance along the circle's circumference for an angle.

Circumference Formula

C = 2πr for a circle.

Unit Circle

Circle with radius 1 centered at the origin.

Sine Function

y-coordinate of point on the unit circle.

Cosine Function

x-coordinate of point on the unit circle.

Tangent Function

Ratio of sine to cosine (tan = sin/cos).

Angle in Radians

Equivalent to arc length for unit circle.

Quarter Circle

Represents 90 degrees or π/2 radians.

Full Revolution

Completing a circle, equals 2π radians.

Reflection Over Y-axis

Inverts x-coordinate of the point on the circle.

Coordinates of P

(cos θ, sin θ) for angle θ on unit circle.

Important Angles

Key angles used in trigonometric calculations.

Angle Measure Example

Finding angle from arc length and radius.

Clockwise Direction

Rotation direction for negative angles.

Counterclockwise Direction

Rotation direction for positive angles.

Symmetry in Angles

Angles in different quadrants share sine/cosine values.

Coordinates of Point P

Point where terminal ray intersects circle.

Standard Position Angle

Angle measured from positive x-axis.

Circle Radius

Distance from center to any point on circle.

Cosine Function

Adjacent side over hypotenuse in right triangle.

Sine Function

Opposite side over hypotenuse in right triangle.

Unit Circle

Circle with radius 1 centered at origin.

Pythagorean Theorem

a² + b² = c² for right triangles.

Symmetry in Unit Circle

Equal angles yield equal sine and cosine values.

Important Angle Values

Common angles: 0, π/6, π/4, π/3, π/2.

Coordinates of Point R

Derived from intersection with line y=-x.

Coordinates of Point S

Derived from unit circle properties.

Angle π/6

Corresponds to 30 degrees on the unit circle.

Angle π/3

Corresponds to 60 degrees on the unit circle.

Angle π/2

Corresponds to 90 degrees on the unit circle.

Pythagorean Theorem

x² + y² = r² for a circle.

Isosceles Triangle

Triangle with two equal sides and angles.

Angle Measure

Degrees or radians representing rotation.

Coordinates

Ordered pairs (x, y) on a plane.

Symmetry of Unit Circle

Coordinates in quadrants mirror each other.

Equilateral Triangle

Triangle with all sides and angles equal.

Sine Function

Ratio of opposite side to hypotenuse.

Cosine Function

Ratio of adjacent side to hypotenuse.

Period

Length of one complete wave cycle.

Transformations of Graphs

Shifts, stretches, or reflections of functions.

Quadrant I

Region where x and y are positive.

Quadrant II

Region where x is negative, y is positive.

Quadrant III

Region where x and y are negative.

Quadrant IV

Region where x is positive, y is negative.

Absolute Maximum Value

Highest point on a graph.

Absolute Minimum Value

Lowest point on a graph.

Even Function

Function symmetric about the y-axis.

Domain of Cosine

All real numbers, (-∞, ∞).

Range of Cosine

Values between -1 and 1, [-1, 1].

Midline of Graph

Horizontal line at y = 0 for cosine.

Midline

Horizontal line at y=0 for sinusoidal functions.

Period

Time taken for one complete cycle, P=2π.

Amplitude

Distance from midline to maximum or minimum.

Frequency

Reciprocal of the period, 1/P.

Sine Function

g(t) = sin(t), oscillates between -1 and 1.

Cosine Function

g(t) = cos(t), similar oscillation as sine.

Domain of Sine

All real numbers: (-∞, ∞).

Range of Sine

Values between -1 and 1: [-1, 1].

Absolute Maximum Value

Highest point of the sine function, 1.

Absolute Minimum Value

Lowest point of the sine function, -1.

Odd Function

Function symmetric about the origin, sin(-t) = -sin(t).

Graph Symmetry

Sine function is symmetric in quadrants.

Transformation

Change from f(t) to g(t) alters graph shape.