Download Review for Quiz 4

Review for Quiz 4
Math 141
page 1
Please note that Quiz 4 will also cover topics covered on Quizzes 1-3 and Exam 1. Please review those topics as
well, even if they do not appear on this document.
1. Convert to radians.
a) 45
b)
120
c) 10
d) 135
e) 270
d)
e)
2. Convert to degrees.
a)
b)
6
c)
9
5
12
3
7
4
3. Use the unit circle to …nd the exact value of each of the following expressions.
7
4
5
d) sin
6
a) sin 120
e) cos ( 90 )
c) tan
b) cos 120
f) cos
5
6
5
3
g) sec
i) tan ( 270 )
j) sec
h) sin
2
5
3
4. The following are all equal to each other, except for one. Which one?
A) cos 150
B) sin ( 60 )
C) cos 210
D) cos ( 120 )
E) sin 300
5. Simplify each of the following expressions. Use exact values and rationalize denominators.
a)
tan 120
tan 45
1 + tan 120 tan 45
b)
sin ( 240 ) cos 135
sin 30 sin 315 tan 300
sin 150 sec 225 tan ( 30 )
c) cot 300 sec 210 tan2 240
6. Which of the following angles are co-terminal? Group co-terminal angles together.
585
135
630
270
225
300
540
90
180
660
495
180
780
540
60
7. Find all angles coterminal with 120 : Express your answer
a) in degrees
b) in radians
c) Find all coterminal angles
such that
500 <
< 500 :
8. True or false?
a) If an angle
is co-terminal with 40 ; then 2 is co-terminal with 80 :
b) If twice an angle 2 is co-terminal with 20 , then
is co-terminal with 10 .
9. a) Compute the exact value of all six trigonometric function values of
if
b) Compute the exact value of all six trigonometric function values of
if
is an acute angle with cos
is an acute angle with tan
3
7
=M
=
10. Simplify each of the following.
a) log3 (tan 60 ) + log2 (sin 45 )
b) sec 30
csc 45 + cot 60
ln (tan 45 )
tan 60
tan 45
1 + tan 60 tan 45
d) tan2 30
tan2 60
c)
11. a) Find the radius if a circle if we know that the length of an arc subtended by a central angle of 85 is 28 cm.
b) Find the radius if a circle if we know that the area of a sector de…ned by a central angle of 35 is 75 cm2 .
Review for Quiz 4
Math 141
page 2
12. Compute the exact value of each of the following.
a) sin 120 cos 225
tan ( 60 )
csc
+ sin2
4
6
3
c) 2 sin 150 cos 240
b) sec
sin
g)
sin
7
4
7
6
tan
+ tan
7
6
9
4
d) sin4 ( 225 ) + cos4 ( 135 )
e) sin 28
cos 62
sin ( 45 ) cos 300
cos ( 45 ) sin 300
f)
tan 225 tan 330
tan
h)
3
1 + tan
tan
3
tan
4
4
13. a) Compute the exact value of sin 30 + sin 30
b) Compute the exact value of sin 60
c) Based on your …ndings, determine whether the following statement is true or false:
For all angles and , sin ( + ) = sin + sin :
d) True or false? For all angles , sin 2 = 2 sin
14. a) Compute the exact value of tan 30 + tan 30
b) Compute the exact value of tan 60
c) Based on your …ndings, determine whether the following statement is true or false:
For all angles and , tan ( + ) = tan + tan .
d) True or false? For all angles , tan 2 = 2 tan
15. Suppose that C is a circle with radius 12 feet.
following.
Compute the exact and approximate values of each of the
a) The length of an arc subtended by a central angle of 42 .
b) The area of a sector determined by a central angle of 42 .
16. a) Find the speed of the sattelite that orbits the Earth above the equator at a height of 1200 miles and
appears at the same point in the sky at all times. Round your answer to the nearest miles per hour. (Assume
that Earth is a sphere with radius 3960 mi)
b*) Find the speed of the sattelite that orbits the Earth above Chicago at a height of 1200 miles and appears
at the same point in the sky at all times. Assume that Earth is a sphere with radius 3960 mi and that the
latitude of Chicago is 42 N . Round your answer to the nearest miles per hour.
17. How fast is the endpoint of the minute pointer moving on my watch if the pointer is 1:2 centimeters long?
Express your answer in centimeters per minute.
18. Berlin, Germany and Copenhagen, Denmark have nearly the same longitude. Use this fact to …nd the distance
between the two cities if the latitude of Berlin is 52:5 N and the latitude of Copenhagen is 55:7 N. Assume
that Earth is a sphere with radius 3960 mi. Round your answer to the nearest mile.
19. Compute the exact and approximate values for the perimeter and area of an 18 sided regular polygon that
was written in a circle with radius 7 centimeters.
20. We are driving away from a tower. First, the angle of elevation from the car to the top of the tower is 52 .
After we drove 300 feet away from the tower, the angle of elevation from the car to the top of the tower is
32 . How tall is the tower?
21. Find the approximate values of the angles in a right triangle where for angle
we know that sin
= 2 cos .
Review for Quiz 4
Math 141
page 3
22. Prove each of the following identities.
a)
1 cos x
sec x 1
=
1 + cos x
sec x + 1
sec x + csc x
d)
= sin x + cos x
tan x + cot x
sin x cos x
+
=1
csc x sec x
b) tan x
c)
csc x sec x 1
2 cos2 x = cot x
23. Consider a circle with radius 10 m. Let P be a point 35 m away from the center of the circle. Compute the
angle formed between the two tangent lines drawn from P to the circle.
24. Compute the area of a triangle with sides 15 m, 15 m, and 22 m.
25. (Enrichment) At noon, the minute and hour hands of the clock exactly overlap.
happen again for the …rst time after noon?
At what time does that
Answers
1. a)
4
2. a) 30
p
3
2
3. a)
b)
2
3
b)
20
c)
c) 75
1
2
b)
d)
18
c)
1
3
4
d)
1
2
d)
e)
540
e) 0
f)
3
2
e) 315
p
3
g) 2
2
h)
1
i) unde ned
j) 2
4. D
5. a)
p
3+2
b)
3
6. Group 1: 135 ; 495 ;
Group 3: 180 ;
c) 2
225 ;
585
180 ; 540 ;
Group 2:
540
60 ; 300 ; 660 ;
Group 4:
780
90 ; 270 ; 630
7. a) 120 + k 360 where k = 0; 1; 1; 2; 2; 3; 3; :::::
2
b)
+ 2k where k = 0; 1; 1; 2; 2; 3; 3; :::::
3
c)
240 ; 120 ; 480
8. True or false?
a) If an angle
True:
is co-terminal with 40 ; then 2 is co-terminal with 80 :
= 40 + k 360
=) 2 = 80 + k 720
b) If twice an angle 2 is co-terminal with 20 , then
False: Let
20 .
= 190 . Clearly 10 and 190 are not co-terminal, but 2 = 380
p
p
3
2 10
2 10
9. a) sin =
cos =
tan =
7
7
3
M
1
b) sin = p
cos = p
tan = M
2
1+M
1 + M2
10. a) 0
b)
1008
11. a)
cm
17
is co-terminal with 10 .
p
3
p
2
c) 2
18: 87390384 cm
p
3
d)
30
b)
7
r
p
p
7 10
7
3 10
csc =
sec =
cot =
20
3
20
p
2
p
1
1+M
sec = 1 + M 2 cot =
csc =
M
M
8
3
42
cm
which is co-terminal with
15: 6701417 cm
Review for Quiz 4
Math 141
12. a)
p
p
3
13. a) 1
b)
6
4
p
b)
3
2
p
2 3
14. a)
3
14
15. a)
ft
5
p
2
5
4
1
2
c)
1
d)
2
e) 0
=
= 30 .
c) False, it clearly failed for
b)
p
3 c) False, it clearly failed for
8: 79646 ft
b)
84
ft
5
=
page 4
f)
p
p
3 2
6
4
g)
p
2
2p
3
3
d) False, it clearly failed for
= 30 .
h) 2
= 30
d) False, it clearly failed for
= 30
52: 778 756 58 ft
mi
mi
b*) 1004
h
h
cm
cm
17. 0:04
0:1256637
min
min
16. a) 1351
18. 221 mi
19. P = 252 sin 10 cm
43: 759341 cm
A = 882 sin 10 cos 10 cm2
150: 830883 cm2
20. 366: 2785 ft
21.
= tan
22. a)
12
63: 434 95
= 90
26: 565 05
sin x cos x
+
=1
csc x sec x
LHS =
b) tan x
csc x sec x 1
=
=
=
sin x cos x
+
=
csc x sec x
sin x
1
sin x
+
cos x
1
cos x
= sin2 x + cos2 x = 1 = RHS
2 cos2 x = cot x
sin x
1
1
1 2 cos2 x
cos x sin x cos x
sin x
1
2 cos2 x
sin x
1
2 cos x
+
=
+
cos x sin x cos x sin x cos x
cos x sin x cos x
sin x
1
2 cos2 x
sin2 x 1 + 2 cos2 x
sin2 x + cos2 x 1 + cos2 x
sin2 x
+
=
=
sin x cos x sin x cos x sin x cos x
sin x cos x
sin x cos x
1 1 + cos2 x
cos2 x
cos x
=
=
= cot x = RHS
sin x cos x
sin x cos x
sin x
LHS = tan x
c)
and
csc x sec x 1
2 cos2 x =
1 cos x
sec x 1
=
1 + cos x
sec x + 1
1
cos x
1 cos x
1
1
sec x 1
1 cos x
cos x
1 cos x
RHS =
= cos x
= cos x cos x = cos x =
=
= LHS
1
1
cos
x
1
+
cos
x
sec x + 1
cos x
1 + cos x
1 + cos x
+1
+
cos x
cos x cos x
cos x
sec x + csc x
d)
= sin x + cos x
tan x + cot x
LHS =
=
sin x + cos x
sin x + cos x
1
1
sin x
cos x
+
+
sec x + csc x
x = cos x sin x = cos x sin x
= cos x sin x = cos x 2sin x sin x cos
2x
1
sin x cos x
tan x + cot x
sin2 x + cos2 x
sin
x
cos
+
+
cos x sin x
cos x sin x
cos x sin x
cos x sin x sin x cos x
sin x + cos x cos x sin x
= sin x + cos x = RHS
cos x sin x
1
p
3
Review for Quiz 4
Math 141
23. 2 sin
1
10
35
p
24. A = 22 26 m2
25. 1
33: 203 1
112: 178 43 m2
1
hour after noon, that is approximately 1 hour, 5 minutes, and 27 seconds after 12:00 noon.
11
Last revised: February 24, 2015
page 5