Angles and Radian Measure Section 4.1

Angle A ray is a part of a line that has only one endpoint and extends forever in the opposite direction. An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side. The endpoint of an angle’s initial side and terminal side form the vertex of the angle.

Standard Position An angle is in standard position if: Its vertex is at the origin of a rectangular coordinate system Its initial side lies along the positive x-axis

Positive/Negative Angles Positive angles are generated by counterclockwise rotation. Thus, angle α is positive. Negative angles are generated by clockwise rotation as you see angle θ in the diagram.

Quadrant Angles An angle is called quadrantal if its terminal side lies on the x-axis or the y-axis. If a standard angle has a terminal side that lies in a quadrant then we say that the angle lies in that quadrant. Angle α lies in quadrant II. Angle θ lies in quadrant III.

Names of Angles

RADIANS Radians Activity

Definition of a Radian One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.

Conversion between Degrees and Radians Use the relationship that π radians = 180° OR Use the relationship that 2π radians = 360°

Examples Convert each angle in degrees to radians 135 −120° −150° 90° 180°

Examples Convert each angle in radians to degrees. 𝜋 2 𝜋 − 𝜋 3 5𝜋 6 2𝜋 3

Drawing Angles in Standard Position

Angles formed by Revolution of Terminal Sides

Angles formed by Revolution of Terminal Sides

Examples Draw and label each angle in standard position. 𝛼= 3𝜋 2 𝛽=2𝜋 𝜃= 7𝜋 4

Coterminal Angles 2π An angle of x ° is coterminal with angles of x °+ k ∙ 360° where k is an integer.

Examples

Examples

Examples

Review (a) (b) (c) (d)

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