Exercises!

Don't wory, it's not an head-ache!
It's very simple, and to make it even simple, ive given you many hints.

Each set of exercises includes first the statements of the exercises, second some hints to solve the exercises, and third the answer to the exercises.


1. Express the following angles in radians. (a). 12 degrees, 28 minutes, that is, 12° 28'. (b). 36° 12'
[To convert degrees to radians, first convert the number of degrees, minutes, and seconds to decimal form. Divide the number of minutes by 60 and add to the number of degrees. So, for example, 12° 28' is 12 + 28/60 which equals 12.467°. Next multiply by and divide by 180 to get the angle in radians.]

2. Reduce the following numbers of radians to degrees, minutes, and seconds. (a). 0.47623. (b). 0.25412
[Conversely, to convert radians to degrees divide by and multiply by 180. So, 0.47623 divided by and multiplied by 180 gives 27.286°. You can convert the fractions of a degree to minutes and seconds as follows. Multiply the fraction by 60 to get the number of minutes. Here, 0.286 times 60 equals 17.16, so the angle could be written as 27° 17.16'. Then take any fraction of a minute that remains and multiply by 60 again to get the number of seconds. Here, 0.16 times 60 equals about 10, so the angle can also be written as 27° 17' 10".]

3. Given the angle a and the radius r, to find the length of the subtending arc. (a). a = 0° 17' 48", r = 6.2935. (b). a = 121° 6' 18", r = 0.2163.
[In order to find the length of the arc, first convert the angle to radians. For 3(a), 0°17'48" is 0.0051778 radians. Then multiply by the radius to find the length of the arc.]

4. Given the length of the arc l and the radius r, to find the angle subtended at the center. (a). l = .16296, r = 12.587. (b). l = 1.3672, r = 1.2978.
[To find the angle, divide by the radius. That gives you the angle in radians]

5. Given the length of the arc l and the angle a which it subtends at the center, to find the radius. (a). a = 0° 44' 30", l = .032592. (b). a = 60° 21' 6", l = .4572.
[As mentioned above, radian measure times radius = arc length, so, using the letters for this problem, ar = l, but a needs to be converted from degree measurement to radian measurement first. So, to find the radius r, first convert the angle a to radians, then divide that into the length l of the arc.]

6.nd the length to the nearest inch of a circular arc of 11 degrees 48.3 minutes if the radius is 3200 feet.
[Arc length equals radius times the angle in radians.]

7. A railroad curve forms a circular arc of 9 degrees 36.7 minutes, the radius to the center line of the track being 2100 feet. If the gauge is 5 feet, find the difference in length of the two rails to the nearest half-inch.
[It helps to draw the figure. The radius to the outer rail is 2102.5 while the radius to the inner rail is 2097.5.]

8.How much does one change latitude by walking due north one mile, assuming the earth to be a sphere of radius 3956 miles?
[You've got a circle of radius 3956 miles and an arc of that circle of length 1 mile. What is the angle in degrees? (The mean radius of the earth was known fairly accurately in 1914. See if you can find out what Eratosthenes thought the radius of the earth was back in the third century B.C.) ]

9.Compute the length in feet of one minute of arc on a great circle of the earth. How long is the length of one second of arc?
[A minute of arc is 1/60 of a degree. Convert to radians. The radius is 3956. What is the length of the arc? ]

10.On a circle of radius 5.782 meters the length of an arc is 1.742 meters. What angle does it subtend at the center?
[Since the length of the arc equals radius times the angle in radians, it follows that the angle in radians equals the length of the arc divided by the radius. It's easy to convert radians to degrees. ]

11.Imagine that the diameter of the balloon is a part of an arc of a circle with you at the center. (It isn't exactly part of the arc, but it's pretty close.) That arc is 50 feet long. You know the angle, so what is the radius of that circle?
[Imagine that the diameter of the balloon is a part of an arc of a circle with you at the center. (It isn't exactly part of the arc, but it's pretty close.) That arc is 50 feet long. You know the angle, so what is the radius of that circle? ]


Answers

1. (a). 0.2176. (b). 0.6318.
2. (a). 27° 17' 10". (b). 14.56° = 14°33.6' = 14°33'36".
3. (a). 0.03259 (b). 2.1137 times 0.2163 equals 0.4572.
4. (a). 0.16296/12.587 = 0.012947 radians = 0° 44' 30". (b). 1.3672/1.2978 = 1.0535 radians = 60.360° = 60° 21.6' = 60° 21' 35".
5. (a). l/a = .032592/.01294 = 2.518. (b). l/a = .4572/1.0533 = .4340.
6. ra = (3200') (0.20604) = 659.31' = 659' 4".
7. The angle a = 0.16776 radians. The difference in the lengths is 2102.5a – 1997.5a which is 5a. Thus, the answer is 0.84 feet, which to the nearest inch is 10 inches.
9. Angle = 1/3956 = 0.0002528 radians = 0.01448° = 0.8690' = 52.14".
10. One minute = 0.0002909 radians. 1.15075 miles = 6076 feet. Therefore one second will correspond to 101.3 feet.
14. a = l/r = 1.742/5.782 = 0.3013 radians = 17.26° = 17°16'.
23. The angle a is 8.5', which is 0.00247 radians. So the radius is r = l/a = 50/0.00247 = 20222' = 3.83 miles, nearly four miles.