Find the Exact Values of the Trigonometric Functions for Quadrailtal Angles
Complete the white portion of the following chart for the Quadrantal Angles:
(}
(}
radians
degrees
00
o"
7r
40°
-
2
7r
31r
I So
0
sinO
cosO
()
I
rl
0
I
170°
-(
2,r
3b[f
0
JL
30°
7T
ti
7T
l
45°
60°
6''
I
~,
\\·~Jrf,<~J
v.v-J~~\QJ
u."Jef/11.l'J
-!
IAV\kf,4r{J J
0
~~Jef~rl
\
ii'3
2
!
--fi
-
-n::
'J--
(i
ff
7
J_
---
;i['i
\h
I
,,:.-----
ff
---
J_
2
0
vtrdr:t~ec
R
\
vi
\
0
uJPh'J
2-D
,
J--
7-
ut'\Jpf,\~
()
r
_I
2
t.
\
'
D
cotO
lA"J(>~:~aj
-I
.,
sec(}
~I\J()R~
1
-0
2
-
cscO
tan(}
-:,,
In order to fill in the information regarding 30°, 45°, 60° angles, recall some geometry.
Exploration 1: Find tt,,e ~xact Valµes of thtr Trigop.pm~tric Func~on~ of . ;
Find the exact values of the six trigonometric functions of
1r
4
=45°
= 45°
1.
1. In'the triangle below, the measure ofthe remaining angle i s : ~
1
I_
2. In the triangle below, the length of the remaining side is: _ _
3. Use the Pythagorean Theorem to find the length of the hypotenuse:
ifi
4. Use the definition of the trigonometric functions to find the exact values ofthe ·six
trigonometric functions:
0
I
fi.
(SC 4~ : {:;_
0
)/V\t.t~
:
L,_~O-
C•~ t\ -
h.\l\ 11~\l;.
lh:.-
' ft
Jj_-~
\
fi.
~~(,,lt)\l;.
Cot ~~
0
::.
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\
Exploration 2: Find the,Exact Values of the Trigonometric Functions of
6
= 30°
and~= 60°
3
Find the exact values of the six trigonometric functions of!!..= 30° and !!.. = 60°
6
3
1. Manipulate the triangle below to find the measure of side a:
~~\/
b
f3
60"
I
a
2. Use the Pythagorean Theorem to find the measure of side b:
Ja~ ,~-=-
@;
3. Use the definition of the trigonometric functions to find the exact values of the six
trigonometric functions: ,{'
3Q0 -- ,_r_J
Csc 3o' ; 2
5 'I'\ 1 00 ::
~I'll"\
I 17
-
t oi 3o0
{L
2
-:o.
to."' 31>°:
Now go back and complete the chart.
se.c
-,
>0
yl
o,
-
Cot 10°:.
6
f3
s~ if=
.
(.
r ,
t>O
-
2.
U c. 60°:
= 30° = 45° = 60°
6
'4
'3
Although we' ve only found the values to COJilp_lete Quadrant 1, we can find the other values
by using the concept of symmetry. Let's explore this.
I. Complete the graphs below.
y
1
1 X
-1
-1
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-1
-
{3
1
J1
'3
1
·
Multiples of
1
I
z:
+"'t\brf -
Exploration 3: Find the Exact Values of the Trigonometric "Functions for Integer
y
tl?!
2. Use the graphs on the previous page to help you fill in the ordered pairs on the unit circle
y
( -l \DJ... )
2, I
( -~ \Ji)
J_
I
2,_
X
( 1, 0 )
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