Name:
Math 1060
Exam 4 Review
For exam 4 you should make sure you know all of the identities from §5.1,
and that you can compute trigonometric functions at special angles.
However, on this exam I will give you the following identities, which you had
to memorize on exam 3:
sin(a + b) = sin a cos b + cos a sin b
cos(a + b) = cos a cos b − sin a sin b
And the following, which you never had to memorize but should understand
how to use:
1
(cos(x − y) − cos(x + y))
2
1
cos(x) cos(y) = (cos(x − y) + cos(x + y))
2
1
sin(x) cos(y) = (sin(x − y) + sin(x + y))
2 x−y
x+y
cos
sin(x) + sin(y) = 2 sin
2
2
x+y
x−y
cos(x) + cos(y) = 2 cos
cos
2
2
sin(x) sin(y) =
The half-angle formulas require a little bit of thought to use:
r
θ
1 + cos θ
=±
sin
2
2
r
θ
1 − cos θ
cos
=±
2
2
You will be able to do all of these problems with just the identities from §5.1–
4, but some problems will be easier if you understand how to use the identities from §5.5.
1. Compute the values of the six trigonometric functions for each of the following angles:
17π
3
(a)
π
3
(c) − π3
(e)
(b)
3π
4
(d) − 7π
6
(f) 120◦
2. Compute
sin(15◦ ) sin(75◦ )
3. Graph the following functions.
Exam 4 Review
(a) y = 4 sin(2x + π) + 2 (b) y = 2 sec
Math 1060
x
2
− π2 − 1 (c) y = tan x − π4 + 1
4. Prove the following identities:
(a) sec2 (θ) tan2 (θ) = sec4 (θ) − sec2 (θ)
(b) cos(π − 2x) = 2 sin2 (x) − 1
5. If u
~ = h2, 1i, v~ = h−1, −3i, compute:
(a) k~
uk
(d) 2~
u + 3~
v
(b) k~
vk
(e) u
~ − 2~
v
(c) k3~
vk
(f) A unit vector in the direction of u
~
6. If u
~ is the vector with magnitude 5 and direction θ =
ponent form?
3π
4 ,
what is it in com-
7. Hilbert walked 7 miles north, then turned east and walked 5 miles. Then
he turned 30◦ to his left and walked 3 miles.
(a) Write down the component form of a vector describing his displacement at the end of his walk.
(b) How far is he from his starting point?
8. If my friend Ug and I are trying to lift up a 100 pound mammoth steak,
and he is pulling up and to the left 30◦ from vertical with 60 pounds of
force, and I’m pulling up and to the right 30◦ from vertical with 60 pounds
of force, will we pulling with enough force to lift up the mammoth steak?
9. I’m driving north in my convertible at 75 miles per hour with the top
down, so I can feel the wind in my hair. But there’s also a wind blowing
from east to west at 20 miles per hour. How fast does the wind feel to me?