Unit 3: Trigonometric and Polar Functions - graphs, equations, and definitions

Periodic phenomenon

An occurrence or relationship that shows a repeating pattern over time or space (e.g., waves, clock hand rotation, daylight hours).

Periodic function

A function that repeats a sequence of output values at fixed input intervals.

Period

The length of one full repetition (one complete cycle) of a periodic function.

Amplitude

The vertical size of an oscillation; for a sinusoid it is the distance from the midline to a maximum/minimum (often |A| in a model).

Concavity

A description of how a graph bends (upward or downward), indicating how the rate of increase/decrease is changing.

Average rate of change

Change in output divided by change in input over an interval: (f(b)−f(a))/(b−a).

Angle (as rotation)

A measure of how much you turn from one ray to another.

Initial side

The starting ray of an angle when describing it by rotation.

Terminal side

The ending ray of an angle after rotating from the initial side.

Standard position

An angle position with vertex at the origin and initial side on the positive x-axis.

Radian

An angle measure where 1 radian subtends an arc length equal to the radius (θ = s/r).

Degree

An angle measure defined by dividing a full circle into 360 equal parts.

Arc length (radians)

The length of an arc cut off by a central angle θ in radians: s = rθ.

Circumference

The distance around a circle: C = 2πr.

Degree–radian conversion

Use 180° = π to convert: θrad = θdeg·(π/180) and θdeg = θrad·(180/π).

Coterminal angles

Angles that share the same terminal side; found by adding/subtracting 360°k or 2πk.

Angle normalization

The process of adding/subtracting multiples of 2π (or 360°) to place an angle in a standard interval such as [0, 2π).

Sector area (radians)

Area of a sector with central angle θ in radians: A = (1/2)r^2θ.

Unit circle

The circle of radius 1 centered at the origin: x^2 + y^2 = 1, used to define trig functions for all real angles.

Pythagorean identity

An identity from the unit circle: cos^2(θ) + sin^2(θ) = 1.

Sine (unit circle definition)

For angle θ, sin(θ) is the y-coordinate of the point where the terminal side hits the unit circle.

Cosine (unit circle definition)

For angle θ, cos(θ) is the x-coordinate of the point where the terminal side hits the unit circle.

Tangent

A trig function defined by tan(θ) = sin(θ)/cos(θ); undefined when cos(θ)=0.

Secant

A reciprocal trig function defined by sec(θ) = 1/cos(θ); undefined when cos(θ)=0.

Cosecant

A reciprocal trig function defined by csc(θ) = 1/sin(θ); undefined when sin(θ)=0.

Cotangent

A trig function defined by cot(θ) = cos(θ)/sin(θ); undefined when sin(θ)=0.

Reference angle

The acute angle between the terminal side of θ and the x-axis, used to evaluate trig values using first-quadrant angles.

Quadrant sign rules

Signs of (sin, cos) by quadrant: I (+,+), II (+,−), III (−,−), IV (−,+).

45-45-90 triangle ratios

Special triangle side ratio 1 : 1 : √2, used for exact trig values (e.g., π/4).

30-60-90 triangle ratios

Special triangle side ratio 1 : √3 : 2, used for exact trig values (e.g., π/6, π/3).

Sine and cosine periodicity

Both repeat every 2π radians: sin(θ+2π)=sin(θ), cos(θ+2π)=cos(θ).

Tangent periodicity

Tangent repeats every π radians: tan(θ+π)=tan(θ).

Tangent/secant domain restriction

tan(θ) and sec(θ) are undefined where cos(θ)=0, i.e., θ = π/2 + kπ.

Cosecant/cotangent domain restriction

csc(θ) and cot(θ) are undefined where sin(θ)=0, i.e., θ = kπ.

Sinusoidal parameter A

In y = A sin(B(x−C)) + D or A cos(B(x−C)) + D, A controls vertical stretch and reflection; amplitude is |A|.

Midline

The horizontal line a sinusoid oscillates around; in y = A sin(B(x−C)) + D, the midline is y = D.

Phase shift

A horizontal translation caused by adding/subtracting a constant inside the trig function (C in B(x−C)).

Frequency

Number of cycles per unit of input; frequency = 1/(period).

Horizontal stretch/compression (parameter B)

In y = A sin(B(x−C)) + D, B changes the period by stretching/compressing horizontally.

Vertical shift (parameter D)

In y = A sin(B(x−C)) + D, D shifts the graph up/down and sets the midline y = D.

Transformed sine/cosine period

For y = A sin(B(x−C)) + D or cos, period = 2π/|B|.

Transformed tangent period

For f(θ)=a tan(b(θ−c))+d, period = π/|b|.

Vertical asymptote (trig graphs)

A vertical line x = constant where a trig function is undefined (e.g., tangent where cos=0; reciprocals where the original function is 0).

Even function

A function with symmetry about the y-axis: f(−x)=f(x); cosine is even.

Odd function

A function with origin symmetry: f(−x)=−f(x); sine is odd.

Sine sum identity

sin(α+β)=sin(α)cos(β)+cos(α)sin(β).

Cosine sum identity

cos(α+β)=cos(α)cos(β)−sin(α)sin(β).

Principal value (inverse trig)

The single angle returned by an inverse trig function from its restricted range (e.g., arcsin in [−π/2, π/2], arccos in [0, π]).

Polar coordinates

A coordinate system using (r, θ) where r is distance from the origin and θ is the angle from the positive x-axis.

Polar-to-Cartesian conversion

Convert (r,θ) to (x,y) using x = r cos(θ) and y = r sin(θ).

Sin Graph

Cosine Graph

Degrees to Radians Conversion

theta = pi/80

Difference between Sine and Cosine graphs

cos Theta = sin (theta + pi/2)

a

amplitude

b

2pi/period

c

phase shift (negative goes right, positive goes left)

d

period

skeleton equation

secant graph

cosecant graph

cotangent graph

Polar functions

(radian, theta)

radian (normal to polar)

r=Square root of x² + y²

Polar graph

r=f(theta)

theta is the input (angle)

Radian is the output

Polar graph

Circle polar graph

when you don´t have an a

Cardioid polar graph

when a=b

limacon polar graph

when b>a

rose polar graph

(something)times(theta)

Polar average rate of change

If the rate is positive, the distance between the point and the origin is increasing - if it is negative, te distance is decreasing