Quiz 6
Math 1060–5
Friday, October 26, 2012
Key
Directions: Show all work for full credit. Clearly indicate all answers. Simplify all mathematical expressions completely. No calculators are allowed.
Formulas
sin(u + v) = sin u cos v + cos u sin v
sin 2u = 2 sin u cos u
sin(u − v) = sin u cos v − cos u sin v
cos 2u = cos2 u − sin2 u
= 2 cos2 u − 1
cos(u + v) = cos u cos v − sin u sin v
cos(u − v) = cos u cos v + sin u sin v
tan u + tan v
tan(u + v) =
1 − tan u tan v
tan u − tan v
tan(u − v) =
1 + tan u tan v
r
1 − cos u
u
sin = ±
2
2
= 1 − 2 sin2 u
2 tan u
tan 2u =
1r
− tan2 u
u
1 + cos u
cos = ±
2
2
u
1 − cos u
sin u
tan =
=
2
sin u
1 + cos u
1. Find the exact value of each of the following: (13 points each)
(a) sin
(b) cos
11π
3π π
11π
(Hint:
=
+ .) (#11 from 5.4)
12
12
4
6
11π
3π π
sin
= sin
+
12
4
6
π
3π
π
3π
cos + cos
sin
= sin
6
√ 4√ 6 √ 4
2
3 − 2 1
=
·
+
·
2
2
2
2
√
√
6− 2
=
4
17π
17π
9π 5π
(Hint:
=
−
.) (#13 from 5.4)
12
12
4
6
17π
9π 5π
cos
= cos
−
12
4
6
9π
5π
9π
5π
= cos
cos
+ sin
sin
4
6
√ 4 √ 6 √
2 − 3
2 1
=
·
+
·
2√ 2√
2 2√
√
− 6+ 2
2− 6
=
=
4
4
(c) tan
π
12
tan
π
π/6
= tan
12
2
1 − cos π6
=
sin π6
√
=
1−
1
2
=2−
3
2
√
3
You could also reach the same answer by using that
π
12
=
π
3
− π4 .
4
2. Let sin u = , where u is an angle in Quadrant II. Find sin 2u. (11 points)
5
Note that cos u = − 35 . You can get this by either using the Pythagorean Theorem or
sketching a triangle. Then,
sin 2u = 2 sin u cos u
4 −3
24
=2· ·
=−
5 5
25