Studied by 13 people
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.
