angles and the unit circle. an angle is in standard position when: 1) the vertex is at the origin....
TRANSCRIPT
Section 13.2
Angles and the Unit Circle
An angle is in standard position when:
1) The vertex is at the origin.
2) One leg is on the positive x – axis.
(This is the initial side.)
3) The second ray moves in the direction of the angle
(This is the terminal side.)
Standard Position Angle (of 60°)
If the movement
from the initial side
to the terminal side
of the angle is
counterclockwise,
then the angle
measures positive.
Reading Angles
+135°
If the movement
from the initial side
to the terminal side
of the angle is
clockwise, then the
angle measures
negative.
Reading Angles
– 225°
Measuring Angles
–315° 240° –110°
1) 2) 3)
Two angles in standard
position that have the same
terminal side are coterminal
angles.
To find a coterminal angle
between 0 ° and 360 ° either
add or subtract 360 ° until you
get the number that you want.
Find the measure of an angle between 0 ° and 360 ° coterminal with each given angle:
4) 575°215 °
5) –356°4 °
6) –210°150 °
7) –180°180 °
Coterminal Angles
The Unit Circle:
1) Is centered at the origin,
2) Has a radius of 1,
3) Has points that relate to
periodic functions.
Normally, the angle measurement
is referred to as θ (theta).
The Unit Circle
1
1
For all values using SOH, CAH, TOA the H value is always 1.
We can use the Pythagorean Theorem to find the rest.
cos θ is the x coordinate. sin θ is the y coordinate. Let’s find sin (60°) and
cos (60°).
On a 30-60-90 triangle the short side is ½ the hypotenuse.
So, cos (60°) = ½. a2 + b2 = c2
(½)2 + b2 = 12
¼ + b2 = 1
b2 = ¾
b = √(¾) = √(3)/2 So, sin (60°) = √(3)/2
Finding Values on the Unit Circle
1
½
√3 2
Continue to find the values on the Unit Circle Find cos 0° and sin 0°
Find cos 30° and sin 30°
Find cos 45° and sin 45°
Find cos 90° and sin 90°
Finding Values on the Unit Circle
1
1
(1, 0)
(0, 1)(½, √3/2)
(√2/2, √2/2)
(√3/2, ½)
These patterns repeat for the right x and y values.
The values can be either positive or negative based on the x and y axes.
Use this information to fill in the worksheets with exact values
All Four Quadrants
Finding Values
Locate the Unit Circle diagram from before.
8) 9)
sin (–60°) = –√(3)/2cos (–60°) = ½
sin (–60°) = –½cos (–60°) = √(3)/2
10) –390°
11) –30°
During this lesson we completed page 708 # 1 – 27 odd.
For more practice, complete the even problems
Extra Practice