4.4 Trigonometric Functions
of Any Angle
!
<" <!
2
3!
! <" <
2
sin " +
cos" !
tan " !
sin ! +
cos! +
tan ! +
sin " !
cos" !
tan " +
sin " !
3!
< " < 2!
cos" +
2
tan " !
!
0 <" <
2
Find the sine, cosine, and tangent of ! if (-3,4)
is a point on the terminal side of ! .
Ex.
4
sin " =
5
(-3,4)
? =5
4
!
-3
3
cos" = !
5
4
tan " = !
3
5
Ex. Given tan " = ! and cos ! > 0 , find
4
sin !
and
sec! .
First, determine what quadrant theta lies in.
Where is tan (-) and cos (+)?
IV
!5
sin # =
" !.7809
41
4
=?
= 41
-5
41
sec" =
! 1.6008
4
Reference Angles
A reference angle is the acute angle formed by the
terminal side of ! and the horizontal axis.
Find the reference angles of 300o, 2.3, and –135o.
1.57
2.3 rad
300o
60o
3.14
3.14
-2.30
.84
45o
-135o
Evaluate the following.
tan (-210o)
1
30o
2
4!
cos
3
240o
2
1
3
=!
3
! 3
1
2
45o
-1
ref. angle
ref. angle
-1
! 3
ref. angle
! 3
11!
csc
4
3!
= csc
4
60o
2
1
!
2
Find the cosine and tangent of theta if…..
1
sin ! =
3
1
3
? =!2 2
and
!
#" #!
2
2 2
cos" = !
3
!1
2
tan " =
=!
4
2 2