Glossary
Amplitude
Half the distance between the maximum and minimum values of a periodic function. It represents the 'height' or intensity of the wave.
Example:
In a sound wave, a higher amplitude corresponds to a louder sound.
Cosecant
The reciprocal of the sine function (csc θ = 1/sin θ). It is one of the three reciprocal trigonometric functions.
Example:
When dealing with the reciprocal of the height of a wave, the cosecant function might appear in the model.
Cosine
One of the primary trigonometric functions, defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle, or the x-coordinate on the unit circle.
Example:
To calculate the horizontal distance a projectile travels given its launch angle and initial velocity, you would typically use cosine.
Cotangent
The reciprocal of the tangent function (cot θ = 1/tan θ). It is one of the three reciprocal trigonometric functions.
Example:
The cotangent of an angle is useful in certain geometric proofs involving slopes and angles.
Equivalent representations
Different forms or expressions that represent the same mathematical value or function. In trigonometry, this often involves using identities to rewrite expressions.
Example:
Knowing that sin(x) = cos(x - π/2) means these are equivalent representations of the same wave.
Inverse trigonometric functions
Functions that determine the angle corresponding to a given trigonometric ratio. For example, arcsin(x) gives the angle whose sine is x.
Example:
If you know the sine of an angle is 0.5, you can use an inverse trigonometric function (arcsin) to find that the angle is 30 degrees or π/6 radians.
Period
The horizontal length of one complete cycle of a periodic function before it begins to repeat itself. For sine and cosine, the standard period is 2π.
Example:
If a pendulum swings back and forth every 2 seconds, its period is 2 seconds.
Periodic phenomena
Patterns or events that repeat over a regular interval of time. Many real-world occurrences, like sound waves or seasons, exhibit this behavior.
Example:
The rise and fall of ocean tides are a classic example of periodic phenomena that can be modeled with trigonometric functions.
Polar coordinates
A two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point (the pole) and an angle from a reference direction (the polar axis).
Example:
Describing the location of a drone relative to its launch point using its distance and bearing would involve polar coordinates.
Polar function graphs
Visual representations of functions defined in polar coordinates, often resulting in unique shapes like cardioids, roses, or spirals.
Example:
The beautiful symmetrical patterns of a rose curve are created by plotting polar function graphs.
Polar functions
Functions defined using polar coordinates (r, θ), where 'r' is the distance from the origin and 'θ' is the angle from the positive x-axis. They are useful for describing circular or spiral patterns.
Example:
To graph the path of a satellite orbiting Earth, a polar function might be more intuitive than rectangular coordinates.
Rates of change
How one quantity changes in relation to another. In polar functions, this often refers to how the radius changes with respect to the angle.
Example:
Analyzing the speed at which a point moves along a spiral defined by a polar function involves calculating its rates of change.
SOH CAH TOA
A mnemonic device used to remember the definitions of the basic trigonometric ratios for a right-angled triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Example:
When solving for an unknown side in a right triangle, remembering SOH CAH TOA helps you choose the correct trigonometric ratio.
Secant
The reciprocal of the cosine function (sec θ = 1/cos θ). It is one of the three reciprocal trigonometric functions.
Example:
In some engineering calculations involving angles, using the secant function can simplify expressions.
Sine
One of the primary trigonometric functions, defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle, or the y-coordinate on the unit circle.
Example:
If you know the angle a ladder makes with the ground and its length, you can use sine to find how high it reaches on a wall.
Sinusoidal functions
Functions that can be expressed in the form y = A sin(Bx + C) + D or y = A cos(Bx + C) + D. They are characterized by their smooth, wave-like oscillations.
Example:
The daily temperature fluctuations throughout a year can often be accurately modeled using sinusoidal functions.
Tangent function
One of the primary trigonometric functions, defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle, or the ratio of sine to cosine.
Example:
To find the angle of elevation of a ramp given its height and horizontal length, you would use the tangent function.
Transformations
Changes applied to a parent function's graph, including shifts (translations), stretches/compressions, and reflections. These alter the position, size, or orientation of the graph.
Example:
Applying transformations to a basic sine wave allows us to model complex real-world data, like varying light intensity over a day.
Trigonometric functions
Functions that relate angles of a right triangle to the ratios of its side lengths, or generalize to the coordinates of points on a unit circle. They are essential for modeling periodic phenomena.
Example:
When analyzing the height of a Ferris wheel rider over time, you would use a trigonometric function like sine or cosine.
Unit circle
A circle with a radius of one unit centered at the origin (0,0) in the Cartesian coordinate system. It is used to define trigonometric functions for all real numbers (angles).
Example:
Finding the exact value of sin(30°) or cos(270°) is made easier by visualizing the coordinates on the unit circle.
Vertical shift
A vertical translation of a function's graph, moving it up or down without changing its shape. In trigonometric functions, it determines the midline.
Example:
If a Ferris wheel's lowest point is 2 meters off the ground, the entire height function will have a vertical shift of 2 meters plus its radius.