PRE-CAL | Trigonometric Angles

Trigonometric Angles

Angle

• the amount of rotation generated when a ray is rotated about its endpoint.

Initial Side

• the initial position of the ray

Terminal Side

• the position of the ray at the end of its rotation

Vertex

• the endpoint of the ray

Positive Angle

• the ray was rotated counterclockwise

Negative Angle

• the ray was rotated clockwise

Angles in Standard Position

• if the vertex is at the origin on a Cartesian plane, and the initial side aligns with the positive part of the x-axis.

Quadrantal Angle

• angle in standard position whose terminal side lies on the x or y axis

Coterminal Angles

• angles that are in the same positions and have the same terminal side but different rotations

Reference Angles (ϴ’)

• the ϴ’ for the given angle is the positive acute angle formed by the terminal side of the given angle and the x-axis

Locations of Reference Angles (based on the location of ϴ)

ϴ is in Quadrant I (0º ≤ ϴ ≤ 90º)

• ϴ = ϴ’

ϴ is in Quadrant II (90º ≤ ϴ ≤ 180º)

• 180º - ϴ = ϴ’

ϴ is in Quadrant III (180º ≤ ϴ ≤ 270º)

• ϴ - 180º = ϴ’

ϴ is in Quadrant IV (270º ≤ ϴ ≤ 360º)

• 360º - ϴ = ϴ’

Radians

• One radian (rad) is the measure of a central angle subtended by the arc equal to the radius of the circle.

Important Angles

• 2π = 360º

• π = 180º

• π/2 = 90º

Converting Radian Measure to Degree Measure

• Multiply the value by 180º/π radians.

Converting Degree Measure to Radian Measure

• Multiply the value by π radians/180º.

Revolution (rev)

• formed when the initial side of an angle rotates completely around the vertex so that the initial and final side coincide.

• 1 rev is equal to 360º or 2π rad

Converting Revolution Measure to Radian Measure

• Multiply the value by 2π rad.

Converting Radian Measure to Revolution Measure

• Divide the value by 2π rad.