Algebra 2/Trig AIIT.13 AIIT.14 AIIT.15 AIIT.16 Angles/Radians Notes
Mrs. Grieser
Name: ________________________________________ Date: _______________ Block: _______
Angles in Standard Position
vertex is on origin of coordinate plane
initial side is on x-axis
terminal side is other ray
counterclockwise motion
In which quadrant does the terminal side lie? ________
In which quadrant does the terminal side of a 310o angle lie? _______
In which quadrant does the terminal side of a -100o angle lie? ________
In which quadrant does the terminal side of a 180o angle lie? ________
In which quadrant does the terminal side of a 440o angle lie? ________
You Try…draw an angle with the given measure in standard position:
a) 240o
b) 405o
c) -65o
Algebra 2/Trig AIIT.13 AIIT.14 AIIT.15 AIIT.16 Angles/Radians Notes
Mrs. Grieser Page 2
Coterminal Angles
Angles in standard position whose terminal sides
coincide
Example: 30o, -330o, and 390o are coterminal angles
To find positive and negative coterminal angles with a
given angle, add or subtract _________ degrees
Example:
a) Find a positive and negative angle coterminal with a
55o angle
b) Find another set!
Another example: Find one positive and one negative angle that are coterminal with:
a) -45o
b) 395o
You Try…
Draw an angle with the given measure in standard position. Then find one positive and
one negative coterminal angle.
a) 485o
b) -75o
Algebra 2/Trig AIIT.13 AIIT.14 AIIT.15 AIIT.16 Angles/Radians Notes
Mrs. Grieser Page 3
Measuring Arcs and Angles
The formula for the circumference of a circle is given by ______________
What is the circumference of a circle with radius 1? ___________
What is the circumference of half a circle with radius 1? ________
What is the circumference of one-quarter of a circle with radius 1? ______
What is the circumference of three-quarters of a circle with radius 1? _______
Using the arc length to represent the opening of an angle may be more intuitive than
dividing a circle into 360 parts! But what if the radius is not 1?
What is the circumference of a circle with radius 2? ___________
What is the circumference of half a circle with radius 2? ________
What is the circumference of one-quarter of a circle with radius 2? ______
What is the circumference of three-quarters of a circle with radius 2? _______
What would these same values be if the radius is 3? ________________________
We don’t want different values for the angle measure is we change the radius!
Compare the arc measures based on radius…
radius
1
2
3
whole circle
2
4
6
2
3
½ circle
¼ circle
3
2
2
¾ circle
3
2
3
9
2
What do we notice? How can we get a consistent value for the angle measure, and why?
Algebra 2/Trig AIIT.13 AIIT.14 AIIT.15 AIIT.16 Angles/Radians Notes
Mrs. Grieser Page 4
Radians
Radians are an alternative means of measuring angles
We define radians…
An angle, θ, in radians is defined by
s
r
where s is the arc length of the arc subtended by the angle, and r is the radius
In a circle with radius 1, θ = s
1 radian = the length of a radius
About how many radians are in a circle?
About how many radians are in half a circle?
Relating Radians and Degrees
Compare radians and degrees to angles we know.
Find the degree angle measures of a circle…
whole circle _______
½ circle _________
¼ circle _________
¾ circle ________
Compare with their radian measures (transfer answer above to table):
degrees radians
whole circle
2
½ circle
¼ circle
2
¾ circle
Notice anything?
3
2
Algebra 2/Trig AIIT.13 AIIT.14 AIIT.15 AIIT.16 Angles/Radians Notes
Mrs. Grieser Page 5
Another way to think about it…
Recall the formula for the length of arc in a circle:
length = s
no
2 r
360o
Example: Find the length s of an arc on a circle with radius 6 that
subtends a 60o central angle.
Solve the formula for the angle…
s
no
2 r
360o
Compare with the formula for an angle in radians
To convert from degrees to radians, we multiply by _____________
Examples: Convert these angles, given in radians, to degrees:
a)
6
b)
3
c) 1
How would we convert from radians to degrees?
To convert from radians to degrees, we multiply by _____________
Examples: Convert these angles, given in degrees, to radians:
a) 45o
b) 210o
c) 150o
Algebra 2/Trig AIIT.13 AIIT.14 AIIT.15 AIIT.16 Angles/Radians Notes
Mrs. Grieser Page 6
You try…convert the degree measure to radians, or the radians measure to degrees:
a) 40o
b) 315o
c) -260o
d) 500o
e)
f)
g) 5
h)
9
-
4
14
15
Special angles
We know…
Special angles in degrees and radians
expanded to a circle:
Write the angles in radians…
Algebra 2/Trig AIIT.13 AIIT.14 AIIT.15 AIIT.16 Angles/Radians Notes
Mrs. Grieser Page 7
Sector Arc Lengths and Area
Sector Arc Length
Recall our formula for θ in radians:
s
r
Solve for s (arc length): ___________
Example:
Find the arc length of a sector with radius 15 inches and a central angle of 60o
Convert angle to radians: 60o = __________
Use formula s = rθ ___________
(give exact answer and estimate to nearest hundredth)
Sector Area
no
We know how to find the area of a sector for angles in degrees: A
360
o
180
To convert to radians, multiply formula by
r2
Area of a sector _____________
Example:
Find the area of a sector with radius 15 inches and a central angle of 60o
Convert angle to radians: 60o = __________
Use formula A
1 2
r ___________ (give exact answer and estimate to nearest hundredth)
2
You try…
Find the exact arc length and area of a sector with the given radius and angle:
r = 5 ft, θ = 75o
Summary:
Arc length of a sector: s = rθ
1 2
Area of a sector: A
r
2