Download Math 104 Midterm 1 Answers

Math 104 Midterm 1 Answers
Problem 1 (5 pts) Identify the following angles as acute, right, obtuse, or straight:
• 39◦ : acute
• 92◦ 50 : obtuse
• 1300 : acute (note: 130 minutes is a little more than 2 degrees)
• π/2 radians: right
• 180◦ : straight
Problem 2 (5 pts) Convert π/5 radians into degrees.
π 180
180
·
=
= 36◦
5 π
5
Problem 3 (5 pts) What is the complement of 12◦ ?
90◦ − 12◦ = 78◦
Problem 4 (5 pts) Find sin(π/4) exactly.
√
2
2
Problem 5 (5 pts) Find cos(150◦ ) exactly.
The reference angle is 180◦ − 150◦ = 30◦ , and this is in the second quadrant where cosine is negative, so
the answer is
√
3
−
2
Problem 6 (5 pts) Consider the angle θ in standard position between the positive x axis and the line going
through (4, 3).
6
r
(4, 3)
θ
The distance r =
So:
√
42 + 32 =
√
-
25 = 5
• sin θ = 3/5
• cos θ = 4/5
• tan θ = 3/4
• cot θ = 4/3
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• sec θ = 5/4
• csc θ = 5/3
Problem 7 (5 pts) Consider the line y = −2x. Let θ be the angle in standard position to this line, in the
second quadrant. Find the following values, exactly:
p
√
A point on this line would be (−1, 2). For this point r = (−1)2 + 22 = 5.
√
• sin θ = 2/ 5
√
• cos θ = −1/ 5
• tan θ = −2
• cot θ = −1/2
√
• sec θ = − 5
√
• csc θ = − 5/2
Problem 8 (5 pts) Suppose θ is an angle in the fourth quadrant, and sin θ = −8/17, and cos θ = 15/17.
Find:
• tan θ = sin θ/ cos θ = −(8/17)/(15/17) = −8/15
• cot θ = 1/ tan θ = −15/8
• sec θ = 1/ cos θ = 17/15
• csc θ = 1/ sin θ = −17/8
Problem 9 (10 pts) You are trying to measure the height of a building. From 100 feet away the angle of
elevation to the top of the building is 70 degrees. Find the height of the building to two significant figures.
The values of the trigonometric functions on 70 degrees is given below. Note that you will not need to use
all of these values.
sin 70◦
cos 70◦
tan 70◦
cot 70◦
sec 70◦
csc 70◦
=
=
=
=
=
=
2
0.940
0.342
2.747
0.364
2.924
1.064
h
◦
70
100 ft
The picture shows a triangle where
tan 70◦ =
h
.
100
Solving for h we have
h = 100 tan 70◦ = 100 · 2.747 = 274.7
Rounding to two significant figures we have 270 feet.
Problem 10 (10 pts) Find the missing lengths and angle in the diagram, to two significant figures (note:
diagram not drawn to scale). The table of trigonometric functions is given below. You will not need to use
all of these values.
A
PP
PP
PP c
PP
b
PP
PP
◦ P
15P
P B
C
100
sin 15◦
cos 15◦
tan 15◦
cot 15◦
sec 15◦
csc 15◦
=
=
=
=
=
=
The missing angle A is 90◦ − 15◦ = 75◦ .
For b we use
tan 15◦ =
0.259
0.966
0.268
3.732
1.035
3.864
b
100
and get
b = 100 tan 15◦ = 100 · 0.268 = 26.8
3
which to two significant figures is 27.
For c we use
sec 15◦ =
c
100
and get
c = 100 sec 15◦ = 100 · 1.035 = 103.5
which to two significant figures is 100.
Problem 11 (10 pts) Find the missing length and angles in the diagram, to three significant figures (note:
diagram not drawn to scale). A table of values for trigonometric functions is given below. You will not need
to use all these values.
A
PP
PP
PP 5
PP
3
PP
PP
PP
P B
C
a
sin−1 (3/5) = 36.869◦
cos−1 (3/5) = 53.130◦
tan−1 (3/5) = 30.963◦
tan−1 (5/3) = 59.036◦
sin−1 (3/8) = 22.024◦
tan−1 (8/3) = 69.444◦
By the Pythagorean theorem,
32 + a2 = 52
and solving, we get
a2 = 52 − 32 = 25 − 9 = 16
This gives us a = ±4, and since a is a length, a = 4. To three significant figures we have 4.00.
To get A we use cos A = 3/5 so that A = cos−1 (3/5) = 53.130◦ . To three significant figures this is 53.1◦ .
To get B we use sin B = 3/5 and so B = sin−1 (3/5) = 36.869◦ . To three significant figures this is 36.9◦ .
Problem 12 (5 pts) If a problem involves the measurements 35.2 and 56.98, and the answer you get on
your calculator says 5.1296523, what answer should you round it to?
Since the least number of significant figures shows up in 35.2 (three significant figures), we need three
significant figures in our answer: 5.13.
Problem 13 (5 pts) Give three different angles θ for which sin θ = 0. Use degrees.
0◦ , 180◦ , 360◦
Also acceptable: −180◦ , −360◦ , 540◦ , 720◦ , . . ..
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Problem 14 (5 pts) Give two angles θ for which tan θ is not defined. Use radians.
π 3π
,
2 2
Also acceptable − π2 , 5π
2 , . . ..
Problem 15 (5 pts) A circle of radius 10 meters is drawn. An angle of 2 radians is drawn at the center.
What is the length of the arc of the circle enclosed by the angle?
Using s = rθ, where r = 10 meters and θ = 2 radians, we get s = 10 · 2 = 20 meters.
Problem 16 (5 pts) In the same problem, what is the area of the region in the circle enclosed by the angle?
Using A = 21 r2 θ = 12 102 · 2 = 100 meters2 .
Problem 17 (5 pts) A merry-go-round has a radius of 10 yards, and rotates once every 40 seconds. If
someone is on the edge of the merry-go-round, how fast are they moving? Leave any factors of π in your
answer.
The angular velocity is 2π radians per 40 seconds, which is
ω=
2π
π
=
radians per second.
40
20
Using v = rω we have
v = 10 ·
π
π
= yards per second.
20
2
Problem 18 (5 pts) If cos θ = −3/5 and tan θ = 4/3, what quadrant is θ in?
Cosine is negative in quadrants II and III; tangent is positive in quadrants I and III. Therefore, θ must
be in quadrant III.
5