Department of Mathematics, University of Wisconsin-Madison
Math 114
Worksheet Sections 7.6, 8.1
1. Find the exact value of each expression.
(a) sin(2θ) and cos
4
3π
θ
if tan θ = , π < θ <
2
3
2
Solution:
4
3π
sin θ
4
Since tan θ = , π < θ <
, one can have
= , and sin θ < 0, cos θ < 0
3
2
cos θ
3
4
3
2
2
Considering sin θ + cos θ = 1, sin θ = − , cos θ = −
5
5
4
3
24
sin(2θ) = 2 sin(θ) cos(θ) = 2(− )(− ) =
5
5
25
r
r
θ
1 − cos θ
4
cos =− (
)=−
2
2
5
θ
θ
π
θ
3π
Note: cos has to be negative because is in the second quadrant by having < <
.
2
2
2
2
4
(b) cos 22.5◦
r
√
◦
1
−
cos
45
2− 2
◦
Solution: cos 22.5 =
=
2
4
◦
Note: cos 22.5 should be positive because 22.5◦ is in the first quadrant.
r
(c) tan
9π
8
v
u
π s
√
u
9π
9π
π u 1 − cos 4
2− 2
Solution: tan
=tan(
− π)=tan =t
π = 2 + √2
8
8
8
1 + cos
4
9π
9π
Note: tan
should be positive because
is in the third quadrant.
8
8
2. Develop a formula for cos(3θ) as a third degree polynomial in terms of cos θ.
Solution:
cos(3θ)=cos(θ + 2θ)=cos θcos 2θ - sin θsin 2θ=cos θ (2cos2 θ - 1) - sin θ(2 sin θ cos θ)
=cos θ (2cos2 θ - 1) - 2sin2 θcos θ=cos θ (2cos2 θ - 1) - 2(1-cos2 θ)cos θ
=(2cos3 θ - cos θ)-2(cos θ - cos3 θ)
=4cos3 θ - 3cos θ
1
3. Establish the identity:
cos θ + sin θ cos θ − sin θ
−
= 2 tan(2θ)
cos θ − sin θ cos θ + sin θ
Solution:
(cos θ + sin θ)(cos θ + sin θ) − (cos θ − sin θ)(cos θ − sin θ)
cos θ + sin θ cos θ − sin θ
−
=
cos θ − sin θ cos θ + sin θ
(cos θ − sin θ)(cos θ + sin θ)
=
(cos2 θ + 2 sin θ cos θ + sin2 θ) − (cos2 θ − 2 sin θ cos θ + sin2 θ)
cos2 θ − sin2 θ
=
4 sin θ cos θ
2 sin 2θ
=
= 2 tan(2θ)
2
2
cos 2θ
cos θ − sin θ
.
Note: There are several ways of establishing identities and the solution here is just giving one way.
4. Solve sin(2θ) = cos θ on the interval 0 ≤ θ < 2π.
Solution:
sin(2θ) = cos θ
2 sin θ cos θ = cos θ
(2 sin θ − 1) cos θ = 0
1
or cos θ = 0
2
π π 5π 3π
θ= , ,
or , 0 ≤ θ < 2π
6 2 6
2
sin θ =
5. Find the exact value of cot 40◦ −
Solution:
sin 50◦
.
sin 40◦
sin 50◦
sin 40◦
◦
cos 40
sin 50◦
=
−
sin 40◦
sin 40◦
◦
◦
sin(90 − 40 ) − sin 50◦
=
sin 40◦
◦
sin 50 − sin 50◦
=
=0
sin 40◦
cot 40◦ −
2
6. Using the right triangle below and knowing that a = 5 and b = 3; find c, A, and B.
b
c
A
B
a
Solution:
p
p
√
a2 + b2 = 52 + 32 = 34
√
−1 a
−1 5
−1 5 34
A = sin
= sin √ = sin
c
34
34
√
b
3
3 34
B = sin−1 = sin−1 √ = sin−1
c
34
34
c=
7. Suppose that you are headed toward a plateau 50 meters high. If the angle of elevation to the top of
the plateau is 20◦ , how far are you from the base of the plateau?
Solution: Suppose the distance between you and the base of the plateau is x, you can have
tan 20◦ =
x=
which means you are
50
x
50
tan
−1
20◦
50
meters away from the base of the plateau.
tan−1 20◦
3