MATH 117
1. Suppose tan ! =
Practice Work
21
with ! in Quadrant III.
2
(a) Find sin ! and cos ! .
(b) Calculate sin(2 ! ) and cos(2 ! ) . In what quadrant is angle 2 ! ?
(c) In what quadrant is angle
!
"!%
"!%
? Calculate sin $ ' and cos $ ' .
#
&
# 2&
2
2
2. Suppose sin ! =
1
with ! in Quadrant II.
2
(a) Find cos ! .
(b) Calculate sin(2 ! ) and cos(2 ! ) . In what quadrant is angle 2 ! ?
(c) In what quadrant is angle
!
"!%
"!%
? Calculate sin $ ' and cos $ ' .
#
&
# 2&
2
2
Solutions
1. Suppose tan ! =
21
with ! in Quadrant III.
2
(a) Find sin ! and cos ! .
tan ! =
21 y
=
2
x
In III, both y and x are negative; so y = ! 21
Then, z =
x 2 + y 2 = 4 + 21 = 5
So sin ! = "
2
21
and cos ! = "
5
5
(b) Calculate sin(2 ! ) and cos(2 ! ) . In what quadrant is angle 2 ! ?
#
21 & # 2 & 4 21
(% " ( =
sin(2 ! ) = 2sin ! cos ! = 2 % "
$
5 '$ 5 '
25
cos(2 ! ) = cos2 ! " sin 2 ! =
4 21
17
"
="
25 25
25
2 ! is in Quadrant II (its sine is + and its cosine is –)
(c) In what quadrant is angle
!
"!%
"!%
? Calculate sin $ ' and cos $ ' .
# 2&
# 2&
2
! is in Quadrant III, so 180º < ! < 270º .
!
!
< 135º ; so is in Quadrant II
2
2
(its sine will be + and its cosine will be –)
Then 90º <
"!%
1 ( cos !
sin $# '& =
=
2
2
1 ( ((2 / 5)
=
2
7/5
7
=
2
10
"!%
1 + cos !
1 + ((2 / 5)
3/5
3
cos $# '& = (
=(
=(
=(
2
2
2
2
10
x = !2
2. Suppose sin ! =
1
with ! in Quadrant II.
2
(a) Find cos ! .
sin ! =
1 y
= . In Quadrant II, y is positive and x is negative. So
2 z
x = ! z2 ! y 2 = ! 4 ! 1 = ! 3
cos ! =
x
3
="
z
2
(b) Calculate sin(2 ! ) and cos(2 ! ) . In what quadrant is angle 2 ! ?
" 1 %"
3%
3
'=(
sin(2 ! ) = 2sin ! cos ! = 2 $# '& $ (
2 # 2 &
2
cos(2 ! ) = cos2 ! " sin 2 ! =
3 1 2 1
" = =
4 4 4 2
2 ! is in Quadrant IV (its sine is – and its cosine is +)
(c) In what quadrant is angle
!
"!%
"!%
? Calculate sin $ ' and cos $ ' .
#
&
# 2&
2
2
! is in Quadrant II, so 90º < ! < 180º .
!
!
< 90º ; so is in Quadrant I
2
2
(its sine will be + and its cosine will be +)
Then 45º <
"!%
1 ( cos !
sin $ ' =
=
# 2&
2
1 ( (( 3 / 2)
=
2
2
3
+
2 2 =
2
2+ 3
4
"!%
1 + cos !
cos $ ' =
=
# 2&
2
1 + (( 3 / 2)
=
2
2
3
(
2 2 =
2
2( 3
4