Chapter 13 Sec 2
Angles and Degree
Measure
Algebra 2 Chapter 13 Sections 2 & 3
Standard Position
•
An angle in standard position has its vertex
at the origin and initial side on the positive
x–axis.
terminal side
initial side
2 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Positively Counterclockwise
•
Angles that have a counterclockwise rotation
have a positive measure.
90º
130
180º
0º
270º
3 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Clockwise means negative
•
Angles that have a clockwise rotation have a
negative measure. – 270º
– 180º
0º
– 130
– 90º
4 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Unit Circle
Now let’s look at angle measures 30, 150, 210, and 330.
150°
180°
(–1, 0)
210°
30°
30º
30º
30º
30º
(1, 0)
330°
They all form a 30°
angle with the x-axis, so
they should all have the
same sine, cosine, and
tangent values…only the
signs will change!
The angle to the nearest
x-axis is called the
reference angle.
All angles with the same
reference angle will have
the same trig values
except for sign5changes.
of 21
Algebra 2 Chapter 13 Sections 2 & 3
Unit Circle
•
•
A unit circle is a circle with radius 1.
If we have an angle between 0o and 90o
in standard position. Let P(x, y) be the
point of intersection. If a perpendicular
segment is drawn we create a right triangle,
where y is opposite θ and x is adjacent to θ.
opp y
sin  
 y
hyp 1
•
adj x
cos  
 x
hyp 1
Right triangles can be formed for angles greater than 90o,
simply use the reference angle.
6 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Radian…still
•
A point P(x, y) is on the unit circle if and only
s
if its distance from the origin is 1.
P(x, y)
•
•
α
The radian measure of an angle is the
length of the corresponding arc on the unit
circle.
C  2r and r  1, thus 360  2 radians
Since

So... 180   radians and 90  radians.
2
7 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Degree/Radian Conversion
Radians 
180

 degree
Degree 

180
 radians
8 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Example 1
a. Change 115o to radian measure in terms of π..

115  115 
180
23

36
7
b. Change 
radian to degree measure.
8
7
7 180



8
8

o
o
 157.5
9 of 21
Algebra 2 Chapter 13 Sections 2 & 3
30° and 45° Radians
•
You will need to know these conversions.
10 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Coterminal Angles
•
•
Coterminal angles are angles that have the same initial
and terminal side, but differ by the number of rotations.
Since one rotation equals 360, the measures of
coterminal angles differ by multiples of 360.
60
60 + 360 =
420
300
300 – 360 =
– 60
11 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Example 2
Find one positive and one negative coterminal angle.
a. 45o
45o + 360o = 405o and 45o – 360 o = –315o
b. 225o
225o + 360o = 585o and 225o – 360 o = –135o
12 of 21
Chapter 13 Sec 3
Trigonometric
Functions
Algebra 2 Chapter 13 Sections 2 & 3
Radius other than 1.
•
•
Suppose we have a hypotenuse with a length
other than 1. For our example we’ll use r as
the length.
In standard position r extends from the Origin to point P(x, y).
14 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Quadrantal Angle
• If a terminal side of an angle coincides with one
of the axes, the angle is called a quadrantal
angle. See below for examples:
• A full rotation around the circle is 360o.
o
Measures more than 360 represent multiple
rotations.
15 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Reference Angles
To find the values of trig functions of angles greater
than 90, you will need to know how to find the
measures of the reference angle.
If θ in nonquadrantal, its reference angle is formed by
the terminal side of the given angle and the x-axis.
16 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Example 1
Find the reference angle for each angle.
a. 312o
Since 312o is between 270o and 360o the terminal
side is in fourth quad. Therefore, 360o – 312o = 48o.
b. –195o
the coterminal angle is 360o – 195o = 165o this
put us in the second quadrant so… 180o – 165o = 15o
17 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Determining Sign
90° (0, 1)
Students
Sine values are positive
180°
(–1, 0)
All
All values are positive
(csc, too)
Take
Tangent values are positive
Calculus
0°/360°
(1, 0)
Cosine values are positive
(sec, too)
(cot, too)
270° (0, –1)
18 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Example 2
Find the values of the six trigonometric functions
for angle θ in standard position if a point with
coordinates (–15, 20) lies on the terminal side.
r
 152  20 2
 625  25
20 4
sin  

25 5
20
4
 15
3
tan  

cos  

 15
3
25
5
25 5
csc  

20 4
25
5
sec  

 15
3
 15
3
cot  

20
4
19 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Example 3
Find the values of the six trigonometric functions
Suppose θ is an angle in standard position whose
4
sec



terminal side lies in the Quadrant III. If
3
find the remaining five trigonometric functions of θ.
r 2  x2  y2
4   3  y 2
2
2
7  y2
y 7
Quad III means
y 7
7
sin   
4
 7
7
tan  

3
3
3
cos  
4
3
3 7
cot  

7
 7
4
4 7
csc   

7
7
20 of 21
Algebra 2 Chapter 13 Sections 2 & 3
Daily Assignment
•
•
Chapter 13 Sections 2 & 3
Study Guide
•
Pg 177
•
•
#4 – 7
Pg 178 – 180 Odd
21 of 21