Transcript unit circle - Coweta County Schools

The
TheUnit
UnitCircle
Circle
• How do we convert angle measures
between degrees and radians?
• How do we find the values of
trigonometric functions on the unit
circle?
HoltMcDougal
Algebra 2Algebra 2
Holt
The Unit Circle
So far, you have measured angles in degrees. You
can also measure angles in radians.
A radian is a unit of angle measure based on arc
length. Recall from geometry that an arc is an
unbroken part of a circle. If a central angle θ in a
circle of radius r is r, then the measure of θ is
defined as 1 radian.
Holt McDougal Algebra 2
The Unit Circle
The circumference of a circle of
radius r is 2 r. Therefore, an angle
representing one complete clockwise
rotation measures 2 radians. You
can use the fact that 2 radians is
equivalent to 360° to convert
between radians and degrees.
Holt McDougal Algebra 2
The Unit Circle
Converting Between Degrees and Radians
Convert each measure from degrees to radians or from
radians to degrees.
.
1. – 60°
3
2.
60
Holt McDougal Algebra 2
o
The Unit Circle
Converting Between Degrees and Radians
Convert each measure from degrees to radians or from
radians to degrees.
.
3. 80°
4
9
4.
20
Holt McDougal Algebra 2
o
The Unit Circle
Converting Between Degrees and Radians
Convert each measure from degrees to radians or from
radians to degrees.
.
5. –36°
5
6. 4 radians
Holt McDougal Algebra 2
The Unit Circle
Reading Math
Angles measured in radians are often not labeled
with the unit. If an angle measure does not have
a degree symbol, you can usually assume that
the angle is measured in radians.
Holt McDougal Algebra 2
The Unit Circle
A unit circle is a circle
with a radius of 1 unit.
For every point P(x, y) on
the unit circle, the value
of r is 1. Therefore, for an
angle θ in the standard
position:
y
tan  
x
Holt McDougal Algebra 2
The Unit Circle
So the coordinates of
P can be written as
(cosθ, sinθ).
The diagram shows
the equivalent
degree and radian
measure of special
angles, as well as
the corresponding xand y-coordinates of
points on the unit
circle.
Holt McDougal Algebra 2
The Unit Circle
Using the Unit Circle to Evaluate Trigonometric Functions
Use the unit circle to find the exact value of each
trigonometric function.
7. cos 225°
The angle passes through the
point
on the unit circle.
Use cos θ = x.
cos 225° = x
Holt McDougal Algebra 2
The Unit Circle
Using the Unit Circle to Evaluate Trigonometric Functions
Use the unit circle to find the exact value of each
trigonometric function.
5
8. tan
6
The angle passes through the
point
on the unit circle.
Use tan θ =
Holt McDougal Algebra 2
.
The Unit Circle
Using the Unit Circle to Evaluate Trigonometric Functions
Use the unit circle to find the exact value of each
trigonometric function.
9. sin 315o
The angle passes through the
point
on the unit circle.
Use sin θ = y.
sin 315° = y
Holt McDougal Algebra 2
The Unit Circle
Using the Unit Circle to Evaluate Trigonometric Functions
Use the unit circle to find the exact value of each
trigonometric function.
10. tan 180o
The angle passes through the
point (–1, 0) on the unit circle.
Use tan θ =
tan 180° =
Holt McDougal Algebra 2
.
The Unit Circle
Using the Unit Circle to Evaluate Trigonometric Functions
Use the unit circle to find the exact value of each
trigonometric function.
4
11. cos
3
The angle passes through the
point
on the unit circle.
Use cos θ = x.
4
1
cos

3
2
Holt McDougal Algebra 2
The Unit Circle
Lesson 10.3 Practice A
Holt McDougal Algebra 2