Glossary
ASTC (All Students Take Calculus)
A mnemonic device that helps determine which trigonometric functions (All, Sine, Tangent, Cosine) are positive in each of the four quadrants of the coordinate plane.
Example:
According to ASTC, in the third quadrant, only the tangent function will have a positive value.
Cosine (cos(θ))
In a right triangle, it's the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. On the unit circle, it corresponds to the x-coordinate.
Example:
When cos(θ) is -1, the angle is 180 degrees, placing the point on the negative x-axis.
Degrees
A common unit of angle measurement where a full circle is divided into 360 equal parts.
Example:
A right angle measures 90 degrees, which is a quarter of a full circle.
Initial Side
The ray of an angle that is fixed along the positive x-axis when the angle is in standard position.
Example:
No matter the angle, its Initial Side always begins pointing to the right along the x-axis.
Inverse Trig Functions (arcsin, arccos, arctan)
Functions that determine the angle when given the ratio of sides (sine, cosine, or tangent). They are used to solve for unknown angles.
Example:
If you know the opposite side is 5 and the hypotenuse is 10, you can use arcsin(5/10) to find the angle is 30 degrees.
Origin (0,0)
The central point on a coordinate plane where the x-axis and y-axis intersect. It is the center of the unit circle.
Example:
All angles on the unit circle are measured starting from the positive x-axis, with their vertex at the Origin (0,0).
Pythagorean Theorem
A fundamental geometric theorem stating that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b), expressed as a² + b² = c².
Example:
The equation of the unit circle, x² + y² = 1, is a direct application of the Pythagorean Theorem where the radius is the hypotenuse.
Radians
A unit of angle measurement where a full circle is 2π radians. It's often preferred in higher-level math and physics.
Example:
An angle of π/2 radians is equivalent to 90 degrees, placing you on the positive y-axis.
SOHCAHTOA
A mnemonic device used to remember the definitions of the three basic trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Example:
Using SOHCAHTOA, you can easily recall that the cosine of an angle is the length of the adjacent side divided by the hypotenuse.
Sine (sin(θ))
In a right triangle, it's the ratio of the length of the side opposite the angle to the length of the hypotenuse. On the unit circle, it corresponds to the y-coordinate.
Example:
If you know sin(θ) = 1/2, you might be working with a 30-degree angle in a right triangle.
Tangent (tan(θ))
In a right triangle, it's the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. On the unit circle, it's the ratio of the y-coordinate to the x-coordinate (y/x).
Example:
If tan(θ) is 1, the angle could be 45 degrees, as both sine and cosine are equal (√2/2).
Terminal Side
The ray of an angle that rotates from the initial side, indicating the angle's measurement and where it stops on the unit circle.
Example:
For a 270-degree angle, the Terminal Side points straight down along the negative y-axis.
Unit Circle
A circle with a radius of 1 unit, centered at the origin (0,0), used as a visual tool to understand angles and trigonometric functions.
Example:
By visualizing the Unit Circle, you can quickly determine that the cosine of 180 degrees is -1.
x-coordinate (as cos(θ))
For any point (x, y) on the unit circle corresponding to an angle θ, the x-coordinate directly represents the cosine value of that angle.
Example:
If the point on the unit circle is (√3/2, 1/2), then the x-coordinate (as cos(θ)) tells us cos(30°) = √3/2.
y-coordinate (as sin(θ))
For any point (x, y) on the unit circle corresponding to an angle θ, the y-coordinate directly represents the sine value of that angle.
Example:
If the point on the unit circle is (0, 1), then the y-coordinate (as sin(θ)) tells us sin(90°) = 1.