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.