Trig Functions of Special
Angles
Objectives
To find exact values for the six trigonometric functions
of special angles
To find decimal approximations for the values of the six
trigonometric functions of any angle
Quadrantal Angles
The first group of special angles are
quadrantal angles.
Quadrantal angles are angles whose terminal side
lies on one of the axes, making it easy to find the
trig functions.
(0, 1)
cos 90 = 0
sin 90 = 1
cos 180 = -1
sin 180 = 0
cos 270 = 0
sin 270 = -1
sin 360 = 0
cos 360 = 1
(1, 0)
(-1, 0)
(0, -1)
Other Special Angles
The other special angles can be found using the trig
version of the Pythagorean Theorem. We will look
at these values in the form of a table.
q (in radians)
q (in degrees)
cos q
sin q
0
p/6 p/4 p/3 p/2 2p/3
0
30°
45°
60°
1
√3/2 √2/2
½
0
½
√2/2 √3/2
3p/4 5p/6
p
90° 120 135 150 180
0
-½
1
√3/2 √2/2
-√2/2 -√3/2
½
0
0
You can also use what we know about reference angles and
their signs in certain quadrants to find the values. Remember,
the reference angle is the acute angle formed by the terminal
side and the axis.
Find each of the following values.
cos 5p/6
terminal side is in quadrant II
The reference angle for 5p/6
in quadrant II is p/6.
cos is x/r.
x in quadrant II is negative.
Cos p/3 is √3/2, but since x is
negative in quadrant II, then
cos 5p/6 = -√3/2.
p
6
5p
6
Find tan(-11p/4)
-11p/4 is in quadrant 3
The reference angle for -11p/4
is p/4.
tan is y/x.
In quadrant 3 y and x are
both negative
Find p/4 in your chart.
Since we don’t know x
and y, we can look at sin
and cos to find x and y.
sin is y/r and cos is x/r
sin p/4 = √2/2 and
cos p/4 = √2/2, so
x = √2 and y = √2/2
p
4
-11p
4
reference
angle
So tan p/4 = √2/√2 or 1.
Find csc 29p/3
29p/3 is coterminal with 5p/3
in quadrant 4
csc is the reciprocal of sin
Find sin p/3 in your chart
sin p/3 = √3/2
p
3
Since csc is the reciprocal of sin
csc p/3 = 2/√3, so csc 29p/3 = 2/√3
Reference angle
When you rationalize the denominator
you get csc 29p/3 = 2√3/3.
Assignment

Page 267 – 268
– # 15 – 35, 42, 44