George W. Ferris' Day Off and Sine Language

Flashcards covering trigonometry in a circular context, angle vocabulary, and motion on a Ferris wheel.

Reference Right Triangles

Right triangles drawn inside circles, used to find the location of points on the circle.

Finding Height Above/Below Center

Draw horizontal line through the center, draw a radius to the point, drop a perpendicular line, label angles between radius & horizontal.

General Ferris Wheel Height Formula

Height = 30 ± 25 * sin(θ), where ± depends on if the height is above or below the center.

Finding Y-coordinate

Y = r * sin(θ), where r is the radius and θ is the angle.

Initial Side

The starting position of an angle.

Terminal Side

The ending position of an angle.

Standard Position

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

Coterminal Angles

Angles in standard position sharing the same terminal side; found by adding/subtracting 360°.

Positive Angles

Angles generated by counterclockwise rotation.

Negative Angles

Angles generated by clockwise rotation.

Reference Angle

A positive acute angle (<90°) formed by the terminal side and the nearest x-axis.

Hypotenuse

The side opposite the right angle in a right triangle.

SOH CAH TOA

Mnemonic for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Elapsed Time

Length of time since passing a specific initial (starting) point.

Angle of Rotation

Degrees between the position of the initial ray and the terminal ray.

Angular Speed

Rate of change of the angle of a rotating object

Period of Rotation

The time it takes for one complete rotation.

Reference Right Triangles

Right triangles drawn inside circles, used to find the location of points on the circle. Helps determine $x$ and $y$ coordinates using trigonometric functions.

Finding Height Above/Below Center

Draw a horizontal line through the center, draw a radius to the point, drop a perpendicular line to the horizontal line, label the angles between the radius and the horizontal line. Can be used to find height $(h)$.
$h = r * sin(θ)$

General Ferris Wheel Height Formula

Height = 30 ± 25 * sin(θ), where ± depends on if the height is above or below the center.
General equation: $h(t) = r * sin(ωt) + k$, where $r$ is the radius, $ω$ is the angular speed, $t$ is time, and $k$ is the vertical shift.

Finding Y-coordinate

Y = r * sin(θ), where r is the radius and θ is the angle.
Used to find vertical displacement from the x-axis.

Initial Side

The starting position of an angle.
Usually along the positive x-axis in standard position.

Terminal Side

The ending position of an angle after rotation.
Determines the angle's quadrant.

Standard Position

An angle with its vertex at the origin and its initial side on the positive x-axis.
Aids in defining trigonometric functions.

Coterminal Angles

Angles in standard position that share the same terminal side; found by adding/subtracting 360° (or $2π$ radians).
$θ' = θ + 360° * n$, where $n$ is an integer.

Positive Angles

Angles generated by counterclockwise rotation from the initial side.
Conventionally considered positive.

Negative Angles

Angles generated by clockwise rotation from the initial side.
Conventionally considered negative.

Reference Angle

A positive acute angle (<90°) formed by the terminal side and the nearest x-axis.
Used to find trigonometric function values in different quadrants.

Hypotenuse

The side opposite the right angle in a right triangle.
Longest side of a right triangle.

SOH CAH TOA

Mnemonic for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

$sin(θ) = \frac{opposite}{hypotenuse}$, $cos(θ) = \frac{adjacent}{hypotenuse}$, $tan(θ) = \frac{opposite}{adjacent}$

Elapsed Time

Length of time since passing a specific initial (starting) point.
Can be used to determine the angle of rotation.

Angle of Rotation

Degrees (or radians) between the position of the initial ray and the terminal ray.

$θ = ω * t$, where $ω$ is the angular speed and $t$ is the time.

Angular Speed

Rate of change of the angle of a rotating object.
Measured in degrees per second or radians per second.

$ω = \frac{Δθ}{Δt}$, where $Δθ$ is the change in angle and $Δt$ is the change in time.

Period of Rotation

The time it takes for one complete rotation.
$T = \frac{2π}{ω}$, where $ω$ is the angular speed.