Transcript TPC Sec 2.1

EQ: How do the x- and y-coordinates of a
point in the Cartesian plane relate to the
legs of a right triangle?
SECTION 2.1
Warm-Up/Activator
Name the quadrants of the Coordinate
(Cartesian) Plane – use drawing on next slide.
 Label the signs of x and y in each quadrant
 If angle measurement travels in the same
direction as the naming of the quadrants,
which direction (clockwise or counterclockwise) is the positive direction?
 Given that the positive end of the x-axis is
the initial side of an angle, and therefore 0˚,
label the corresponding angles on the other
three ends of the x- and y-axis.

Drawing for the Warm-up/Activator
Coterminal Angles

Two angles in standard position are
coterminal if they have the same
__________ ______.
Example 2
Determine whether the following pairs of
angles are coterminal.
 a)   120,   180 b)   20,   740

Your Turn 2
Determine whether the following pairs of
angles are coterminal.
 a)   240,   120 b)   20,   380

Example 3
Determine the angle of the smallest
possible positive measure that is
coterminal with each of the following
angles.
 a) 830˚
b) -520˚

Your Turn 3
Determine the angle of the smallest
possible positive measure that is
coterminal with each of the following
angles.
 a) 900˚
b) -430˚

How do angles in quadrant II, III and IV
relate to angles in quadrant I?
REFERENCE ANGLES
Vocabulary
Terminal Side: the rotating ray of an
angle
 Quadrantal Angles: an angle whose
terminal side lies along the x- or y-axis.
 Reference angle: acute angle formed by
the terminal side and the x-axis

90°
I
II
+
Terminal side
0°
180°
-
III
IV
270°
360°
Reference Angle
To find the reference
angle when the angle is in
Quadrant II, subtract the
angle from 180°.
Angle
Θ = 115°
180° – 115° = 65°
Θ = 225°
Reference Angle
To find the reference
angle when the angle is in
Quadrant III, subtract
180° from the angle.
225° – 180° = 45°
Θ = 330°
To find the reference
angle when the angle is in
Quadrant IV, subtract the
angle from 360°.
Reference Angle
360° – 330° = 30°
Reference Angle
Θ = -150°
360° + – 150° = 210°
210° – 180° = 30°
Example 1

Sketch the following angles in standard position.
State the quadrant in which (or axis on which) the
terminal side lies. Also state the reference angle.

a) -90˚
b) 210˚
Your Turn 1

Sketch the following angles in standard position.
State the quadrant in which (or axis on which) the
terminal side lies. Also state the reference angle.

a) -300˚
b) 135˚
Why do we analyze the values of the special angles in all
quadrants of the unit circle?
ANALYZING VALUES
OF THE UNIT CIRCLE
Special Triangles
Sin 30
Cos 30
60°
Tan 30
1
2
Sin 60
30°
3
Cos 60
Tan 60
Special Triangles
Sin 45
45°
Cos 45
1
2
Tan 45
45°
1
The unit circle is a circle with a
radius of one
P(cosϑ, sinϑ)
P (x,y)
1
y
ϑ
x
sin  
y
y
1
x
cos    x
1
Values of the quadranal angles: What are quadranal angles?
90°
The angles whose terminal sides are on the axes.
(0,1)
Continue on the top circle:
1
-+
180°
(-1,0)
P(cosΘ, sinΘ)
++
Θ
1
--
0°
(1,0)
+270°
(0,-1)
90°
 3 1


 2 ,2


 3 1


 2 ,2


150°
30°
180°
0°
360°
30
210°
330°
 3 1 


 2 , 2 


  3 1 


 2 , 2 


270°
90°
 2 2


 2 , 2 


 2 2

,
45° 

 2 2 
135°
45
180°
0°
360°
315°  2  2 
,
  2  2  225°


 2 , 2 


 2

270°
2 
90°
 1 3 
 ,

 2 2 


120°
60°
60
180°
 1  3 
 ,

 2
2 

0°
360°
300°
240°
270°
1 3
 ,

2 2 


1  3
 ,

2 2 


90°
120°
60°
45°
135°
150°
30°
60
45
180°
0°
360°
30
210°
330°
225°
315°
300°
240°
270°
Unit Circle
Using the Unit Circle to find exact
sin and cos values
Go back to Examples 2 & 3 and determine
the exact sin and cos values for the angles.