Angles in a Unit Circle
● Objectives: The lesson aims to help students illustrate the unit circle, understand the
relationship between linear and angular measures of a central angle, convert between
degree and radian measures, and illustrate angles in standard position to find coterminal
angles.
● Introduction to Trigonometry: Trigonometry is derived from the Greek words "trigon"
(triangle) and "metron" (measure)4.
● Definition of an Angle: In geometry, an angle is the union of two non-collinear rays with
a common endpoint (vertex). One ray is the initial side, and the other is the terminal side.
● Standard Position: An angle is in standard position if its vertex is at the origin and its
initial side lies on the positive x-axis.
○ Counter-clockwise rotation forms a positive angle7.
○ Clockwise rotation forms a negative angle.
● The Unit Circle:
○ A circle centered at the origin with a radius of one unit.
○ It shows the placement of degrees and radians in the coordinate plane10.
○ Used to define radians as a unit of measure for angles, based on the circumference
of the unit circle11.
● Radian Measure:
○ Also defined as the ratio between the length of the arc (s) intercepted by a central
angle and the length of the radius (r):
θ(inradian)=rs
12.
○ One radian (1 rad) is the measure of a central angle where the intercepted arc has
the same length as the radius (both equal to 1)13.
○ Examples:
■ If radius (r) = 40 cm and
θ = 1/2 radian, the arc length (s) = 20 cm14.
