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Algebra III Academic
Trigonometry
Angles in Degrees and Radians, Coterminal Angles and Reference Angles (Day 4)
Warm-Up
Solve the right triangles that follow. (Find all side lengths and all angles)
Example 1:
A
a
7
Example 2:
x
B
A
5
9
C
7
Example 3:
B
A = 30°
C
Example 4: A = 45°
A
A
C
5
B
C
16
B
For the problems that follow, sketch the given angle  in standard position. Then, find one positive and one
negative coterminal angle. In which quadrant does the terminal side of the angle land?
1)
q = 500°
2)
  700
3)
  960
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
Quadrant: _____
Quadrant: _____
Quadrant: _____
+
-
+
-
+
-
Algebra III Academic
Trigonometry: Angles and Degree Measure (Day 4)
Objective:
To use degrees to measure angles and become familiar with the unit circle.
Review coterminal angles and calculate reference angles.
Write angles in different forms.
Reference Angle: ____________________________________________________________________
Illustrations and How to Find Reference Angles…
Quadrant I
Quadrant II
r

r
Quadrant III

Quadrant IV

r
r
Steps for finding reference angles.
1. Given angle must be within one rotation of the circle 0    360. If the given angle is not, find a
positive coterminal angle.
2. Now, find the reference angle:
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
Sketch the following angles and determine the reference angle.
a.  = 57°
b.  = 302°
c.  = 133°
d.  = 254°
e.  = -227°
f.  = -184°
g.  = 4897°
h.  = 220°
Unit Circle
In a unit circle the radius = 1
Circumference: _____________
Area: _____________________
Central Angle   central angle  : begins at the side oriented towards positive x and sweeps in a counterclockwise direction (positive angles), or a clockwise direction (negative angles).
Ways to measure and write angles:
1.
2.
Equivalent Expressions and Solving for Radians or Solving for Degrees
1 trip around a circle in degrees
 360

traveling around the circumference s 2
360s  2
s

180

360
s 2
360s

2
180s


Converting from Degrees to Radians: Angle in degrees 
Converting from Radians to Degrees: Angle in radians 


 Angle in Radians
180
180

 Angle in Degrees
Convert the following angles to degrees or radians.
1) 45°
5)
2p
9
2) 270°
6)
3p
5
3) - 150°
7) -
3p
4
4) 95°
8) -85°
Algebra III Academic
Trigonometry: Angles and Degree Measure (Day 4)
Homework
Convert the following from degrees to radians or vice versa.
7) 225°
11)
2
7
8) 315°
12)
3
4
9) 150°
13) 
7
4
10) 75°
14) -65°
For each of the follow questions, find one positive and one negative coterminal angle. Then, draw the angle in
standard position and find the measure of the reference angle.
15) 87°
qc+ = _____
16) 562°
qc+ = _____
17) - 336°
qc+ = _____
18) - 45°
qc+ = _____
qc = _____
qc = _____
qc = _____
qc = _____
qr = _____
qr = _____
qr = _____
qr = _____
Quadrant: _____
Quadrant: _____
Quadrant: _____
Quadrant: _____
-
-
-
-
Sketch the following angles and determine the reference angle.
a.  = - 57°
b.  = - 77°
c.  = - 133°
d.  = - 254°
Sketch the following angles and determine the reference angle.
e.  = 227°
f.  = 184°
g.  = - 4897°
h.  = - 220°
Algebra III/Trigonometry Quiz Review
Solve the following using trigonometry.
1. A lighthouse keeper observes that there is a 3 angle of depression between the horizontal and the line of
sight to a ship. If the keeper is 19 m above the water, how far is the ship from shore?
2. Suppose you have been assigned the job of measuring the height of the local water tower. Climbing makes
you dizzy, so you decide to do the whole job at ground level. From a point 47.3 meters from the base of the
tower, you find that you must look up at an angle of 53° to see the top of the tower. How high is the tower?
3. Standing across the street 50 feet from a building, the angle to the top of the building is 40°. An antenna sits
on the front edge of the roof of the building. The angle to the top of the antenna is 52°. How tall is the
building? How tall is the antenna itself, not including the height of the building?
4. An observer 80 feet above the surface of the water measures an angle of depression of 42° to a distant ship.
How many miles is the ship from the base of the lighthouse?
5. A student looks out of a second-story school window and sees the top of the school flagpole at an angle of
elevation of 22. The student is 18 feet above the ground and 50 feet from the flagpole. Find the height of
the flagpole
6. From a point on a cliff 75 feet above water level, an observer can see a ship. The angle of depression to the
ship is 4°. How far is the ship from the base of the cliff?
Solve the given right triangles (find all angles and all side lengths). Use your knowledge of special right
triangles and leave all side lengths in simplest radical form.
7.
60
[------ 17 ------]
Find the missing side lengths. Use the properties of special right triangles and write your answers in radical
form!
8.
9.
10.
10 cm
45
30
14 in
36 ft
60
11.
12.
13.
14.
30
15
6
30
30
12 cm
Convert the following to degrees or radians.
14. 78°
15.
4
9
16. 388°
17.
5
3
Sketch the angle, then find its reference angle. Also find one positive and one negative coterminal angle.
1. 345°
2. -35°
3. 48°
4. 215°
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qr = _____
qr = _____
qr = _____
qr = _____
5. 76°
6. 94°
7. 137°
8. 259°
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qr = _____
qr = _____
qr = _____
qr = _____
11. -186°
12. 5481°
+
-
+
-
9. 54°
+
-
+
-
10. -303°
+
-
+
-
+
-
+
-
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qc = _____
qr = _____
qr = _____
qr = _____
qr = _____
+
-
+
-
+
-
+
-