Download Problem Set 6 - Marta Hidegkuti

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Answers
Problem Set 6
Math 141
Use the button on your calculator instead of typing in 3:14: Present exact values (preferred) or
round approximations to three decimal places.
1. Use the unit square to …nd the exact values for each of the following
p
2
1
(a) sin 45 = p =
2
2
(b) tan 45 = 1
(c) sin2 45 + cos2 45 = 1
2. The radius of a circle is 8 m.
8
m = 2: 793 m
9
Solution: The circumference of the circle is C = 2 r = 2 (8 m) = 16 m. This is the arc
belonging to the central angle of 360 . The rest is just a ratio problem.
angle arc
360
C
20
x
(a) Find the length of an arc subtended by a central angle of 20 .
x
20
=
C
360
=)
x=C
20
360
=
C
16 m
8
=
=
18
18
9
m ' 2: 793 m
32
m2 = 11: 170 m2
9
Solution: The area of the circle is A = r2 = (8 m)2 = 64 m2 . This is the area of the
sector belonging to the central angle of 360 . The rest is just a ratio problem.
angle area of sector
360
A
20
x
(b) Find the area of a sector subtended by a central angle of 20 .
x
20
=
A
360
=)
x=A
3. Find the polar coordinates for (20; 15).
20
360
=
A
64 m2
32
=
=
18
18
9
m2 ' 11: 170 m2
(25; 0:644 rad) = (25; 36:870 )
4. Find the rectangular coordinates for (3; 41 ).
(2: 264 ; 1: 968 )
5. Convert 15 to radians. Round your answer to three decimal places. 0:262
6. Convert 1 radian to degrees. 57: 296
7. Let ABC be a right triangle with sides a; b; and c: (c is the hypotenuse.) Express sin2
in terms of a; b; and c. 1
+ cos2
2
8. Find the angle that the straight line y = x + 5 forms with the positive half of the x-axis.
3
33: 690 = 0:588 rad
c copyright Hidegkuti, Powell, 2008
page 2
Answers
Problem Set 6
Math 141
9. Find the exact value for all six trigonometric function of , based on the following picture.
sin
=
12
;
13
cos
=
5
;
12
tan
=
12
;
5
csc
=
13
;
12
sec
=
12
;
5
cot
=
5
12
10. Find the radius of a circle if we know that a central angle of 70 subtends an arc of 12 in.
216
in = 9: 822 in
7
11. We are driving toward a tower. The angle of elevation is 28 . Then we drive 550 ft toward the
tower. Now the angle of elevation is 40 . How tall is the tower? 798:290 ft
Solution: Based on the picture below, we can write a system of linear equations.
tan 28
tan 40
H
x + 550
H
=
x
=
In this system, there are two unknowns, H and x. We will use substitution to solve the system. It
is a bit unusual, but common in trigonometry to symbolically carry the trigonometric expressions
and only to substitute approximate values into them in the last line.
tan 28
tan 40
H
=)
x + 550
H
=
=)
x
=
tan 28 (x + 550)
x tan 28 + 550 tan 28
550 tan 28
550 tan 28
550 tan 28
tan 40
tan 28
H = tan 40 x = tan 40
c copyright Hidegkuti, Powell, 2008
550 tan 28
tan 40
tan 28
=
=
=
=
H = tan 28 (x + 550)
H = tan 40 x
tan 40 x
x tan 40
x tan 40
x (tan 40
x tan 28
tan 28 )
= x
=
550 tan 40 tan 28
= 798: 289 78 ( ft)
tan 40
tan 28
Problem Set 6
Math 141
12. Find the exact value of 2 sin
4
cos
4
page 3
Answers
= 1
13. Green Bay, WI and Mobile, AL have approximately the same longitude. The radius of the earth
is approximately 3960 miles. The latitude of Green Bay is 44:5 and that of Mobile is 30: 7 : Find
the distance to the nearest mile between the two cities. 303: 6 mi = 954 mi
Solution: A circle has radius 3960 miles. Find the length of the arc subtended by a central angle
of 44:5
30:7 = 13: 8 .
14. The picture below shows a straight pyramid with a square base. The sides of the base are 10 in
long. The other sides are 15 in long.
p
(a) Find the height of a triangluar edge. 10 2 in = 14: 142 in
p
(b) Use part a) to …nd the height of the pyramid. 5 7 in = 13: 229 in
15. Find the side of a regular polygon with 10 sides that is written in a circle of radius 28.
17: 305 units
16. An object is traveling on a circular orbit of radius 4 meters. It completes 10 cycles in one minute.
4 m
m
Find the speed of the object. Express your answer in meter per second.
= 4: 189
3 s
s
c copyright Hidegkuti, Powell, 2008