Practice Final Exam
Math 1060 Fall 2010
Dec 14th 2010
Name and Unid: |
{z
}
by writing my name i swear by the honor code
Read all of the following information before starting the exam:
• Show all work, clearly and in order, if you want to get full credit. I reserve the right to
take off points if I cannot see how you arrived at your answer (even if your final answer is
correct).
• Justify your answers algebraically whenever possible to ensure full credit. Sketch all relevant graphs and explain all relevant mathematics.
• Circle or otherwise indicate your final answers.
• You MAY NOT use a calculator or notes. You may use the formulas provided.
• This test has 10 problems and is worth 200 points, plus some extra credit at the end. It
is your responsibility to make sure that you have all of the pages!
• Good luck!
Some Basic Identities:
sec(x) =
1
cos(x)
csc(x) =
1
sin(x)
cot(x) =
1
tan(x)
tan(x) =
sin(x)
cos(x)
Pythagorean Identities:
cos2 (x) + sin2 (x) = 1
1 − sin2 (x) = cos2 (x)
1 − cos2 (x) = sin2 (x)
1 + tan2 (x) = sec2 (x)
1 + cot2 (x) = csc2 (x)
Addition and Subtraction Fromulas:
sin(α ± β) = sin(α) cos(β) ± cos(α) sin(β)
cos(α ± β) = cos(α) cos(β) ∓ sin(α) cos(β)
tan(α ± β) =
tan(α) ± tan(β)
1 ∓ tan(α) tan(β)
Half Angle Identities:
√
θ
sin( ) = ±
2
θ
cos( ) = ±
2
√
1 − cos(θ)
2
1 + cos(θ)
2
Double Angle Formulas:
sin(2θ) = 2 sin(θ) cos(θ)
cos(2θ) = cos2 (θ) − sin2 (θ)
= 2 cos2 (θ) − 1
= 1 − 2 sin2 (θ)
tan(2θ) =
2 tan(θ)
1 − tan2 (θ)
Power Reducing Formulas:
1 + cos(2x)
2
1 − cos(2x)
sin2 (x) =
2
cos2 (x) =
Law of Sines:
sin(A)
sin(B)
sin(C)
=
=
a
b
c
Law of Cosines:
a2 = b2 + c2 − 2bc cos(A)
b2 = a2 + c2 − 2ac cos(B)
c2 = a2 + b2 − 2ab cos(C)
cos(A) =
b2 + c2 − a2
2bc
cos(B) =
a2 + c2 − b2
2ac
cos(C) =
a2 + b2 − c2
2ab
Area:
1
1
1
Area = bc sin(A) = ac sin(B) = ab sin(C)
2
2
2
√
a+b+c
Area = s(s − a)(s − b)(s − c), s =
2
Vectors: Magnitude of a vector:
√
k~uk =
u21 + u22
cos(θ) =
~u · ~v
k~ukk~v k
Angle between two vectors:
Projection of ~u onto ~v
(
proj~v (~u) =
)
~u · ~v
~v
k~v k2
Complex numbers: If z1 = r1 (cos(θ1 ) + i sin(θ1 )) and z2 = r2 (cos(θ2 ) + i sin(θ2 )) then.
z1 · z2 = r1 r2 (cos(θ1 + θ2 ) + i sin(θ1 + θ2 ))
z1
r1
= (cos(θ1 − θ2 ) + i sin(θ1 − θ2 ))
z2
r2
(
(z1 )n = (r1 )n (cos(nθ1 ) + i sin(nθ1 ))
(z1 )1/n = (r1 )1/n cos
(
θ1 + 2πk
n
)
(
+ i sin
θ1 + 2πk
n
))
, k = 0, 1, . . . , (n − 1)
1.
(20 points) Radians and Degrees
a. (10 pts)
Convert from degrees to radians: 1200 =
b. (10 pts) Convert from radians to degrees:
2.
(20 points) Evaluate the following
a. (5 pts)
arccos(cos( π2 )) =
b. (5 pts)
cos(arcsin(0)) =
c. (5 pts)
sin(arctan(1)) =
d. (5 pts)
tan(arcsin( −1
2 )) =
90π
18
=
3.
(20 points) Using Triangles and Trigonometric Functions
a. (10 pts)
Given that cos(θ) = − 45 for
π
2
< θ < π find sin(θ) and sec(θ).
b. (10 pts) Use a triangle to rewrite sin(arctan( xy )) as an algebraic expression involving
no trigonometric functions only x0 s and y 0 s.
4.
(20 points) Graphs of Trogonometric Functions and Their Inverses.
Below are the graphs of sin(x), cos(x), tan(x), arcsin(x), arccos(x), arctan(x) in no particular order. On each figure write which function the graph represents and state the domain and range
of that function.
K
0.5
p
K
0.25
1.0
10
0.5
5
p
0
0.25
x
p
0.5
p
K
0.5
p
K
0.25
p
0
K
K
K
K
0.5
10
0.25
p
0.5
0.75
p
0.5
p
0.25
p
p
5
1.0
K
p
K
10
1.0
K
0.5
0
0.5
p
1.0
0.5
p
0.25
p
0.5
0.25
p
K
0
0
5
K
0.25
K
0.5
5
10
0.25
p
0.5
p
0.75
p
p
K
1.0
K
0
0.5
p
K
K
p
K
K
0.5
1.0
0.25
0.5
p
p
0.5
1.0
0.5
1.0
5.
(20 points) Graphs of Trigonometric Functions
a. (5 pts)
What is the period of the function f (x) = sin(4(x − π/4))?
b. (15 pts) Sketch the graph of y = sin(4(x − π/4)). Label both axis. Label at least two
points on each axis.
2
1
Kp
2
K
1.5
p
Kp
K
0.5
p
0
0.5
p
p
x
1.5
p
2
p
K
1
K
2
6.
(20 points) Using Trigonometric Identities
In the figure below the angles a are equal. Use the double angle formula for tan to solve for the
missing length z.
z=?
y=3
2a
a
a
x=4
7.
(20 points) Story Problem
A biologist wants to know the width w of a river in order to properly set instruments for
studying the pollutants in the water. She notices a very large boulder directly across the river
from where she is standing. She proceeds to walk 200ft downstream and sights the boulder.
From the sighting it is determined that the angle between the river bank and the line from her
new position to the boulder is 390 . How wide is the river?
8.
(20 points) Solving Trigonometric Equations
For each of the following find
√ the set of all x’s that solve the equation:
a. (5 pts)
cos(x) = 22
b. (5 pts)
sin2 (x) −
c. (10 pts)
sin2 (x) cos2 (x) + cos4 (x) = 0
3
4
=0
9.
(20 points) Law of Sines and Cosines
a. (10 pts)
Use the Law of Sines to solve for side b for the given triangle.
C
a=40
b
60
A
0
45
0
B
c
b. (10 pts) Use the Law of Cosines to solve for the angle B for the given triangle.
C
a=7
b=2
A
B
c=10
10.
(20 points) Vectors
Let ~u =< 1, 2 > and ~v =< 1, −3 >
a. (5 pts)
Sketch ~u and ~v
b. (5 pts)
Find 3~u + 2~v and sketch the result.
c. (5 pts)
Find ~u · ~v .
d. (5 pts)
Find the angle between ~u and ~v leave your answer interms of arccos.
11.
(20 points) Coordinate Systems for the 2-dimensional Plane.
a. (10 pts)
A( point) in polar coordinates is given. Convert the point to rectangular
coordinates. (r, θ) = 2, −π
2
b. (10 pts)
A( point in
rectangular coordinates is given. Convert the point to polar
√ )
coordinates. (x, y) = − 3, 1
12.
(5 points) Complex Numbers(BONUS!)
Consider the complex number z = 4 + 2i.
a. (1 pt)
Sketch z.
b. (1 pt)
Find the absolute value of z.
c. (3 pts)
easier?) .
Find z 200 . (Hint: Is there is different way you could write z to make this
Scrap Page
(please do not remove this page from the test packet)