Glossary
ASTC Rule
A mnemonic ('All Students Take Calculus') used to remember which trigonometric functions (All, Sine, Tangent, Cosine) are positive in each of the four quadrants of the coordinate plane.
Example:
Using the ASTC Rule, you know that if an angle is in Quadrant III, only its tangent value will be positive.
Constant Ratios
The principle that for a given angle, the trigonometric ratios (sine, cosine, tangent) remain the same regardless of the size of the right triangle, as long as the angle is consistent.
Example:
Whether you have a small 30-60-90 triangle or a large one, the constant ratios mean sin(30°) will always be 1/2.
Cosine (cos θ)
For an angle θ on the unit circle, cosine is defined as the x-coordinate of Point P. In a right triangle, it's the ratio of the adjacent side to the hypotenuse.
Example:
At 60° on the unit circle, the cosine value is 0.5, representing the x-coordinate.
Degrees
A unit of angular measurement, where a full circle is divided into 360 equal parts. It is commonly used alongside radians for angle measurement.
Example:
A right angle measures 90 degrees, which is a quarter of a full rotation.
Multiple Revolutions
Angles that extend beyond a single full rotation (360° or 2π radians), indicating that the terminal ray has completed one or more full circles before landing on its final position.
Example:
An angle of 720° represents two full multiple revolutions around the unit circle, ending at the same position as 0°.
Negative Angles
Angles measured in a clockwise direction from the positive x-axis. They represent a rotation in the opposite direction of positive angles.
Example:
A clockwise rotation of 45° from the positive x-axis results in a -45° negative angle.
Point P
The intersection point of the terminal ray of an angle in standard position and the unit circle. Its coordinates are (cosθ, sinθ).
Example:
If an angle is π/2 radians, Point P on the unit circle will be at (0,1).
Positive Angles
Angles measured in a counterclockwise direction from the positive x-axis. This is the standard direction for angle measurement.
Example:
Rotating counterclockwise from the positive x-axis to the positive y-axis forms a +90° positive angle.
Radians
A unit of angular measurement, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius. A full circle is 2π radians.
Example:
Converting 180 degrees to radians gives you π, which is half a circle's rotation.
Reflex Angles
Angles that measure greater than 180° but less than 360°. They represent a rotation past a straight line but not a full circle.
Example:
An angle of 270° is a reflex angle because it's larger than 180° but less than 360°.
SOHCAHTOA
A mnemonic acronym used to remember the definitions of the three basic trigonometric ratios in a right-angled triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Example:
When solving for a missing side in a right triangle, remembering SOHCAHTOA helps you choose the correct trigonometric function.
Sine (sin θ)
For an angle θ on the unit circle, sine is defined as the y-coordinate of Point P. In a right triangle, it's the ratio of the opposite side to the hypotenuse.
Example:
If you're at 30° on the unit circle, the sine value is 0.5, which is the y-coordinate.
Standard Position
An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. Positive angles are measured counterclockwise.
Example:
An angle of 120° is in standard position if its initial side is on the positive x-axis and its terminal side is in Quadrant II.
Tangent (tan θ)
For an angle θ on the unit circle, tangent is defined as the ratio of the y-coordinate to the x-coordinate of Point P (sinθ/cosθ). In a right triangle, it's the ratio of the opposite side to the adjacent side.
Example:
The tangent of 45° is 1, because sin(45°)/cos(45°) = (√2/2)/(√2/2) = 1.
Terminal Ray
The side of an angle in standard position that rotates from the initial side. Its position determines the angle's measure.
Example:
For an angle of 180°, the terminal ray lies along the negative x-axis.
Unit Circle
A circle with a radius of 1 unit, centered at the origin (0,0) on the coordinate plane, used as a fundamental tool for understanding trigonometric functions.
Example:
When you're trying to visualize sin(90°), you can easily locate the point (0,1) on the unit circle.