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Algebra II/Trig Honors
Unit 9 Day 7: Define General Angles and Use Radian Measures
Objective: Draw and use general angles that may be measured in radians or degrees
Definitions:
Angle in Standard Position – vertex of the angle is at the origin
Initial Side – fixed ray on the positive x-axis
Terminal Side – rotated ray somewhere else in the coordinate plane
Label Quadrants – I, II, III, IV
The measure of an angle is positive if the rotation of
its terminal side is counterclockwise
The measure of an angle is negative if the rotation
of its terminal side is clockwise
The terminal side of an angle can make more than one complete rotation; this will make the angle greater than
360 .
Example 1: Draw an angle with the given measure in standard position.
a. 240
b. 500
c.  50
Practice: Draw an angle with the given measure in standard position.
a. 65
b. 740
c.
100
Co-terminal Angles – angles that have the same initial side and the same terminal side. They differ by sign and
number of rotations.
Co-terminal Angles are found by adding and subtracting multiples of 360 ® qc = q g ± 360°
*POSITIVE co-terminal angles must be POSITIVE *NEGATIVE co-terminal angles must be NEGATIVE
Example 2: Find one positive angle and one negative angle that are co-terminal with the following angles:
a.  45
b. 395
Practice: Find one positive angle and one negative angle that are co-terminal with the following angles:
a. 230
b. 150
c. 425
Radians – another way to measure angles, in terms of  rather than degrees. Radians represent the length of an
arc cut from a circle of radius 1. Each angle will have a radian measure in terms of  .
The circumference of a circle is 2 r, let’s treat r = 1, and say a full circle is 2 radians. That makes half of a
circle equal to  .
360  2
180  
You can convert angles between degrees and radians using the formulas below.
Degrees to Radians
Angle ×
p
180°
Radians to Degrees
Angle ×
180°
p
Example 3: Convert between degrees and radians
a. Convert 125 to radians
b. Convert 

12
to degrees
Sector of a Circle – region of a circle that is bounded by two radii and an arc of the circle (it’s a fraction of the
circle’s area)
Central Angle  of a sector is the angle formed by the two radii (its vertex is at the center of the circle)
Arc Length is the distance around the circle bounded by the two radii (it’s a fraction of the circle’s
circumference)
Know the formulas or APPLY what you know from geometry!
Formulas: *****Both formulas must have  measured in RADIANS
Arc Length: s  r
1 2
r 
2
Example 4: Solve a multi-step problem
A softball field forms a sector with the dimensions shown. (Angle is 90 , length of a side is 180 feet). Find the
length of the outfield fence and the area of the field.
Area of a Sector: A 
Children at a day camp are playing a game on a circular field. The shaded sector in the figure is called the “safe
zone,” and is marked off by rope along its outer edges. Find the length of the rope and the area of the safe zone.
Assignment: Worksheet