An angle is formed by two rays with a common vertex.
Initial Side: One ray of the angle.
Terminal Side: The other ray of the angle.
Direction and Amount of Rotation: Shown by a curved arrow from initial to terminal side.
Counterclockwise Rotation: Positive angle.
Clockwise Rotation: Negative angle.
Standard Position of an Angle
Vertex: Located at the origin of a rectangular coordinate system.
Initial Side: Coincides with the positive x-axis.
Example:
Angle Theta (θ) in quadrant 1 with counterclockwise rotation is positive.
Angle Theta (θ) in quadrant 3 with clockwise rotation is negative.
Sketching Angles in Standard Position
Axes Representation in Degrees:
Positive x-axis = 0°
Positive y-axis = 90°
Negative x-axis = 180°
Negative y-axis = 270°
Positive x-axis (full circle) = 360°
Example Angles
30 Degrees:
Quadrant 1, counterclockwise rotation.
Sketch with initial side and vertex same for all angles.
Negative 120 Degrees:
Requires clockwise rotation.
Axes representation for negative rotation:
Negative y-axis = -90°
Negative x-axis = -180°
Positive y-axis = -270°
Positive x-axis (full circle) = -360°
Located in quadrant 3 because -120° is between -90° and -180°.
Identify rotation with a curved arrow clockwise.
450 Degrees:
Recognize that 450° is greater than 360°.
Start with vertex and initial side at the positive x-axis, rotate counterclockwise.
Mark the positive y-axis as 450° (360° + 90°).
Terminal side on positive y-axis.
Identify full revolution plus additional rotation with a curved arrow covering 360° and additional 90° (total 450°).
Note: For angles greater or less than one full circle (360°), mark extra revolutions and directions accordingly. For negative angles like -450°, use clockwise rotation.