4.1: Radian and Degree Measure
Objectives:
•To use radian measure of an angle
•To convert angle measures back and forth between
radians and degrees
•To find coterminal angle
We are going to look at angles on the coordinate
plane…
 An angle is determined by rotating a ray about its
endpoint
 Starting position: Initial side (does not move)
 Ending position: Terminal side (side that rotates)
 Standard Position: vertex at the origin; initial side
coincides with the positive x-axis
 Positive Angle: rotates counterclockwise (CCW)
 Negative Angle: rotates clockwise (CW)
Negative Angle
Positive Angles
1 complete rotation: 360⁰
Angles are labeled with Greek letters:
α (alpha), β (beta), and θ (theta)
Angles that have the same initial and terminal
sides are called coterminal angles
RADIAN MEASURE (just another unit of measure!)
 Two ways to measure an angle: radians and degrees
 For radians, use the central angle of a circle
r
s=r
•s= arc length intercepted by angle
•One radian is the measure of a
central angle, Ѳ, that intercepts an
arc, s, equal to the length of the
radius, r
• One complete rotation of a circle = 360°
• Circumference of a circle: 2r
• The arc of a full circle = circumference
s= 2r
Since s= r, one full rotation in radians= 2 =360 °
2  6.28 , so just over 6 radians in a circle
2  360 (1 revolution)
½ a revolution =
¼ a revolution
1/6 a revolution=
1/8 a revolution=
Quadrant 2

2
 
  
3
2
Quadrant 3
Quadrant 1
0  

2
3
   2
2
Quadrant 4
Coterminal angles: same initial side and terminal side
Name a negative coterminal angle:

2


3
2

You can find an angle that is coterminal to a given angle by
adding or subtracting 2
Find a positive and negative coterminal angle:

1.
6

2. 
3
2
3.
3
7
4.
2
Degree Measure
360  2
180  
So………
1 

rad
180
180
1rad 
deg

Converting between degrees and radians:

1. Degrees →radians: multiply degrees by 180
180
2. Radians → degrees: multiply radians by

Convert to Radians:
1. 320°
2. 45 °
3. -135 °
4. 270 °
5. 540 °
Convert to Radians:
1. 

2
2 .3
6
3.
5
5
4.
4
Sketching Angles in Standard Position: Vertex is
at origin, start at 0°
1.  3
4
2. 60°
Sketch the angle
3. 13
6
4.3 Right Triangle Trigonometry
Objectives:
• Evaluate trigonometric functions of acute
angles
• Evaluate trig functions with a calculator
• Use trig functions to model and solve real
life problems
Right Triangle Trigonometry
Side
opposite
θ
hypotenuse
θ
Side adjacent to θ
Using the lengths of these 3 sides, we form six
ratios that define the six trigonometric
functions of the acute angle θ.
sine
cosine
tangent
cosecant
secant
cotangent
*notice each pair has a “co”
Trigonometric Functions
• Let θ be an acute angle of a right triangle.
opp
sin  
hyp
adj
cos  
hyp
opp
tan  
adj
RECIPROCALS
csc  
hyp
opp
hyp
sec  
adj
adj
cot  
opp
Evaluating Trig Functions
• Use the triangle to find the exact values of the
six trig functions of θ.
hypotenuse
4
θ
3
Special Right Triangles
45-45-90
30-60-90
45°
60°
2
1
2
1
45°
1
30°
3
Evaluating Trig Functions for 45°
• Find the exact value of sin 45°, cos 45°, and
tan 45°
Evaluating Trig Functions for 30° and
60°
• Find the exact values of sin60°, cos 60°,
sin 30°, cos 30°
60°
30°
Sine, Cosine, and Tangent of Special
Angles

1
sin 30  sin 
6 2
0
p
1
sin 45 = sin =
4
2
0

3
sin 60  sin 
3
2
0

3
cos 30  cos 
6
2
0
p
1
cos 45 = cos =
4
2
0
cos 600  cos

3

1
2
tan 300  tan

tan 450  tan
tan 60  tan
0
6


4

3
1
3
1
 3
Trig Identities
• Reciprocal Identities
1
sin  
csc 
csc  
1
sin 
1
cos  
sec 
sec  
1
cos 
1
tan  
cot 
cot  
1
tan 
Trig Identities (cont)
• Quotient Identities
sin 
tan  
cos 
cot  
cos 
sin 
Evaluating Using the Calculator
• sin 63°
• tan (36°)
• sec (5°)
Applications of Right Triangle
Trigonometry
• Angle of elevation: the angle from the
horizontal upward to the object
• Angle of depression: the angle from the
horizontal downward to the object
Word Problems
• A surveyor is standing 50 feet from the base of
a large tree. The surveyor measure the angle
of elevation to the top of the tree as 71.5°.
How tall is the tree?
• You are 200 yards from a river. Rather than
walk directly to the river, you walk 400 yards
along a straight path to the river’s edge. Find
the acute angle θ between this path and the
river’s edge.
• Find the length c of the skateboard ramp.