Download 10.3B The Unit Circle IV III II I

10.3B The Unit Circle
Objectives:
F.TF.2: Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of angles
traversed counterclockwise around the unit circle.
F.TF.1: Understand radian measure of an angle as the length of the arc on the unit circle
subtended by the angle.
F.TF.4: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric
functions.
For the Board: You will be able to convert angle measures between degrees and radians and find the
values of trigonometric functions on the unit circle.
Anticipatory Set:
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.
For an angle θ in standard position:
sin θ = y/r = y/1 = y
cos θ = x/r = x/1 = x
tan θ = y/x
(x, y) = (cos θ, sin θ)
Instruction:
The Unit Circle
 1 3
 ,

 2 2 




2π
2 2


,
 2 2  3π 3


120°
4

135°
3 1  5π


,
 2 2
6


150°
(-1, 0) π 180°
(0, 1)
π
2
90°
II I
1 3
 ,

2 2 


π
 2 2


,
3
π  2 2 
60°
4
45°
π  3 , 1 
 2 2
30° 6


0°
0 (1, 0)
III IV

3 1


 2 , 2 


7π 210°
6
5π 225°

2
2 4
4π 240°


 2 ,- 2 
3


 1
3
  ,
 2 2 


330°
11π 

315°
6 
7π
270° 300°
5π 4  2
3π
 2 ,3

2
1
3
 ,
(0, -1)
2 2 


3 1
, 
2
2
2

2 
Complete the Unit Circle Handout:
1. Label the points with their radian measures:
0, π/2, π, 3π/2
π/4, 3π/4, 5π/4, 7π/4
π/6, π/3, 2π/3, 5π/6, 7π/6, 4π/3, 5π/3, 11π/6
2. Label the points with their degree measures:
0°, 90°, 180°, 360°
45°, 135°, 225°, 315°
30°, 60°, 120°, 150°, 210°, 240°, 300°, 330°
3. Label the points on the x and y axis: (1, 0), (0, 1), (-1, 0), (0, -1)
4. Recall: cos θ = x and sin θ = y
Therefore each point (x, y) is (cos θ, sin θ)
Use special right triangles to find the first quadrant ordered pairs
5. Use symmetry and the sign chart for the quadrants
to complete the other ordered pairs.
(--, +)
(+, +)
(--, --)
(+, --)
Instruction:
Open the book to page 708 and read example 2.
Example: Use the unit circle to find the exact value of each trigonometric function.
a. cos 225°
b. tan 5π/6
cos θ = x so
tan θ= y/x so
1
2
5π
1
3 1
2
1
cos 225° = 
tan
 2  
 

2
6
2
2
3 2
3
3

2
1
3
3


=
3
3 3
White Board Activity:
Practice: Use the unit circle to find the exact value of each trigonometric function.
a. sin 315°
b. tan 180°
-½
y/x = 0/-1 = 0
c. cos 4π/3
-½
Open the book to page 708 and read example 3.
Example: Use a reference angle to find the exact value of the sine, cosine, and tangent of 330°.
1. Determine the measure of the reference angle: 30°
2. Use quadrant I of the unit circle to find the sine, cosine, and tangent of 30°:
1
3
1
3 1 2
1
3
3
sin 30° = ½
cos 30° =
tan 30 = 2  
 



2
2 3
3
3 2 2
3 3
2
3. Adjust the sign based on the quadrant: 330° is quadrant IV.
3
3
sin 30° = - ½
cos 30° =
tan 30° = 
2
3
White Board Activity:
Practice: Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle.
11
a. 270°
b.
c. -30°
6
11
sin 270° = -1
sin
=-½
sin -30° = - ½
6
3
3
11
cos 270° = 0
cos
=
cos -30° =
2
2
6
3
3
11
tan 270° = undefined
tan
=
tan -30° = 
3
3
6
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 711 prob. 10 – 17, 27 – 34.
For a Grade:
Text: pg. 711 prob. 28, 32.