Download 10.2B Angles of Rotation

10.2B Angles of Rotation
Objective:
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.
For the Board: You will be able to draw angles in standard position and determine the values of the
trigonometric functions for an angle in standard position.
Anticipatory Set:
For an angle θ in standard position, the reference angle is the positive acute angle formed by the
terminal side of θ and the x-axis. This is easiest done by sketching the location of the angle.
Open the book to page 701 and read example 3.
Example: Find the measure of the reference angle for each given angle.
a. θ = 135°
b. θ = -135°
c. θ = 325°
Reference angle
θ
θ
θ
180° – 135° = 45°
180° – 135° = 45°
White Board Activity:
Practice: Find the measure of the reference angle for each given angle.
a. θ = 105°
b. θ = -115°
75°
65°
360° – 325° = 35°
c. θ = 310°
50°
Instruction:
To determine the value of the trigonometric functions for an angle θ in standard position.
1. Select a point P with coordinates (x, y) on the terminal side of the angle.
2. The distance r from point P to the origin is given by x 2  y 2 .
(This is the Pythagorean Theorem: x2 + y2 = r2)
3.
Sine
Cosine
Tangent
y
x
y
sin θ =
cos θ =
tan θ =
r
r
x
Note: x and/or y could be negative or positive based on the
quadrant.
Recall: SOHCAHTOA based on θ'.
P(x, y)
r
θ
y
θ'
x
Open the book to page 702 and read example 4.
Example: P(-6, 9) is a point on the terminal side of θ in standard position.
Find the exact value of the six trigonometric functions for θ.
x = -6 and y = 9 so r =
sin θ =
9

3 13
6
cos θ = 
3 13
9 3
tan θ =  
6 2
(6) 2  9 2  36  81  117  3 13
3 13
13

13
3
csc θ =
2 13
13
13
2
sec θ = 
cot θ = 
2
3
White Board Activity:
Practice: P(-3, 6) is a point on the terminal side of θ in standard position.
Find the exact value of the six trigonometric functions for θ.
x = -3 and y = 6 so r =
sin θ =
6
3 5
cos θ = 
3

2 5
5

3 5
6
tan θ =   2
3
(3) 2  6 2  9  36  45  3 5
csc θ =
5
5
sec θ =  5
cot θ = 
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 703 – 704 prob. 10 – 25, 34 – 49, 52 – 55.
For a Grade:
Text: pg. 703 prob. 36, 42.
5
2
1
2