Glossary
Cosine (cos)
For an angle in standard position, cosine is the ratio of the x-coordinate of a point on the terminal ray to the radius (x/r), or simply the x-coordinate on the unit circle.
Example:
If a point on the unit circle is (0.6, 0.8), then the cosine of the angle is 0.6.
Coterminal Angles
Angles that share the same terminal ray, differing by integer multiples of 360° (or 2π radians).
Example:
An angle of 30 degrees and an angle of 390 degrees are coterminal angles because they end at the same position after different rotations.
Graphs of Trig Functions
Visual representations of trigonometric functions, showing their periodic nature, amplitude, and phase shifts.
Example:
Understanding the wave-like shape of the graph of the sine function helps predict its behavior over time.
Initial Ray
The starting side of an angle in standard position, always located along the positive x-axis.
Example:
For any angle drawn on a coordinate plane, the initial ray always points to the right along the x-axis.
Negative Angles
Angles measured by rotating clockwise from the positive x-axis.
Example:
An angle of -45 degrees is a negative angle, indicating a clockwise rotation from the positive x-axis.
Periodicity (Sine/Cosine)
The property of sine and cosine functions to repeat their values every 2π radians (or 360 degrees).
Example:
The graph of sin(x) completes one full wave and then repeats, demonstrating its periodicity of 2π.
Periodicity (Tangent)
The property of the tangent function to repeat its values every π radians (or 180 degrees).
Example:
Unlike sine and cosine, the periodicity of tangent is π, meaning its graph repeats twice as fast.
Positive Angles
Angles measured by rotating counterclockwise from the positive x-axis.
Example:
An angle of 270 degrees is a positive angle because you rotate counterclockwise from the x-axis to reach its terminal side.
Radian-Degree Conversions
The process of converting angle measurements between radians and degrees using the relationship 180° = π radians.
Example:
To convert 90 degrees to radians, you'd multiply 90 by π/180, resulting in π/2 radians, which is a radian-degree conversion.
Radians
A unit of angle measurement defined by the ratio of the arc length subtended by the angle to the radius of the circle.
Example:
A full circle measures 2π radians, which is approximately 6.28 units of arc length for a radius of 1.
Range (Sine/Cosine)
The set of all possible output values for the sine and cosine functions, which is always [-1, 1].
Example:
No matter the angle, the value of sin(θ) will always fall within the range of -1 to 1.
Range (Tangent)
The set of all possible output values for the tangent function, which is all real numbers (-∞, ∞).
Example:
Because tangent can represent any slope, its range extends infinitely in both positive and negative directions.
Sine (sin)
For an angle in standard position, sine is the ratio of the y-coordinate of a point on the terminal ray to the radius (y/r), or simply the y-coordinate on the unit circle.
Example:
If a point on the unit circle is (0.6, 0.8), then the sine of the angle is 0.8.
Standard Position
An angle is in standard position when its vertex is at the origin (0,0) and its initial ray lies along the positive x-axis.
Example:
When you graph an angle of 90 degrees, its starting line is on the positive x-axis, making it in standard position.
Tangent (tan)
For an angle in standard position, tangent is the ratio of the y-coordinate to the x-coordinate (y/x), which also represents the slope of the terminal ray.
Example:
If the terminal ray passes through (3, 4), the tangent of the angle is 4/3, representing the slope.
Terminal Ray
The ending side of an angle, formed by rotating from the initial ray.
Example:
After rotating 180 degrees counterclockwise from the positive x-axis, the terminal ray of the angle would lie along the negative x-axis.
Trigonometric Identities
Equations involving trigonometric functions that are true for all values of the variables for which the functions are defined.
Example:
The identity sin²(θ) + cos²(θ) = 1 is a fundamental trigonometric identity used to simplify expressions.
Unit Circle
A circle with a radius of 1 unit, centered at the origin, used to define trigonometric functions.
Example:
When finding the sine of an angle, you can visualize the y-coordinate of the point where the terminal ray intersects the unit circle.