Warm up
• Fill in the blank unit circle
Lesson 11 Nice Angles
Objective: To evaluate trig values of the
nice angles.
Nice Angles
30o
45o
60o
90o
π/6
π/4
π/3
π/2
• We should now be able to find
the point on the unit circle that
corresponds to multiples of
any of these angles (positive or
negative).
The Unit Circle
©Carolyn C. Wheater, 2000
• The length of its legs are the xand y-coordinates of the
chosen point.
• Applying the definitions of the
trigonometric ratios to this
triangle gives
x
cos    x
1
 is the
angle of
rotation
y
sin( )   y
1
4
1
y
x
• So sin θ= y/1 and
(x, y) =
(cos θ, sin θ) cos θ= x/1
1
θ
x
y
y
• This works for any
angle on the unit
circle.
The Unit Circle
• The coordinates of the chosen point are the cosine
and sine of the angle .
– This provides a way to define functions sin() and cos()
for all real numbers .
x
cos( )   x
1
©Carolyn C. Wheater, 2000
y
sin( )   y
1
– The other trigonometric functions can be defined from
these.
6
©Carolyn C. Wheater, 2000
Trigonometric Functions
sin( )  y
1
csc  
y
cos( )  x
1
sec  
x
y
tan  
x
x
cot  
y
7
 is the
angle of
rotation
1
y
x
Find the trig function (not in decimal form)
• Sin 240o =
 3
2
• Cos 240o =
1
2
• Sin 420o =
3
2
• Cos -60o =
1
2
All Star Trig Class
• Use the phrase “All Star Trig Class” to remember
the signs of the trig functions in different
quadrants.
Star
©Carolyn C. Wheater, 2000
Sine is
positive
Trig
Tan is
positive
All
All functions
are positive
Class
Cos is
positive
9
Other Trig Functions
• Use the relationships that you already know
between sin, cos and the other trig functions to be
able to find the other trig ratios of any of the special
angles.
• Sec(45o) =
1
cos 45
=
1
2
2
=
2
2 2

 2
2
2
Find using the unit circle
• Cot 60
• Csc 120
• Sec 300
Graphing sin, cos and tan functions
•
•
•
•
•
To graph y = sin x:
[Y=]
Type in sin x
[ZOOM] [7]
[GRAPH]
• If your mode is in degrees the x axis will read
degrees,
• If your mode is in radians the x axis will read radians.
Sine Graph
Cosine Graph
Tangent Graph