Radian+and+Degree+Measure

=Radians and Degrees=

by: Richard Forbes, Kara Kenney


media type="custom" key="525897" media type="file" key="Radian vs. Degree.m4a" = __Radians and Degrees song lyrics__ = (just in case you couldn't make them out, read them here with all their genius exposed) 3.14 (dot dot dot) is the number Of Radians in a semicircular In a full circle, radians double Pi turns into 2 pi just to make trouble Radians radians used by Canadians Degrees degrees make me sneeze (achoo) Degrees divide a circle up odd 360 degrees in a circle made of cod Divide that in two, and you find a semicircle 180 degrees that encircles Radians radians – there everythings arcadian Degrees degrees they are quite a breeze (woosh) Switching back and forth from one to the other Will quickly make you dread another Calculators make it easy But then math teachers couldn’t keep us busy Radians radians- nothing to do with circadian Degrees degrees they are the bee’s knees (bzzz) To get from radians to degrees you use 180 over pi if there are a lot you will want to sigh For radians to degrees, you use the reciprocal Doing this a lot is also very dull. Radians radians used by Canadians Degrees degrees make me sneeze (achoo)

There are two different ways of measuring angles. One uses degrees, and one uses radians. Both methods are described here.

=Radians:= Radians are a process of measuring angles. Instead of degrees, one uses the length of the radius. The angle that radius covers has been found to be a little more than 3. π radians make up a semicircle. All angles are based off this information. 360 degrees = 2 π or 0 π 180 degrees = π 90 degrees = π /2 45 degrees = π /4 30 degrees =π /6 etc.

= Degrees: = Degrees don't use a ratio, but are instead a fraction of the circle. One degree is 1/360 of a revolution around the vertex. A full revolution is 360°, half of a revolution is 180°, and a quarter of a revolution is 90°.

The simplest way to convert radians to degrees is by multiplying the radian measure by 180/π. The simplest way to convert degrees to radians is by multiplying the degree measure by π /180.

=Finding Arc Length, Linear Speed, and Angular Speed:= An angle can also be used to find the length of an arc. When using radians, the formula s = rӨ can be used to find the length of the arc. "s" is the length of the arc, "r" is the radius of the circle, and Ө is the angle in radians. This formula will not work with an angle in degrees, so the angle should be changed into degrees before this forumla is used.

The arc length can also be found by using the formula s = 2π r.

After the arc length has been found, it can be used to find linear and angular speed. Linear speed can be found with the formula: arc length/time, where "arc length" is the length of the arc that is passed through. Angular speed can be found with the formula: central angle/time, where "central angle" is the angle that is passed through.

=__Problems:__=

Answers: 45° & -315°, π/4 & -7π/4

Answers: 210° & -150°, 7π /6 & -5π /6

Change the following radian angles into degrees: a) 3<span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /2 b) 7<span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /3 c) <span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /5 d) 5<span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /6

Answers: a) 270° b) 420° c) 36° d) 150°

Change the following degree angles into radians: a) 30° b) 90° c) 145° d) 240°

Answers: a) <span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /6 b) <span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /2 c) 29<span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /36 d) 4<span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /3

Find the complement and supplement of the following angles: a) <span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /3 b) <span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /12 c) 36° d) 23°

Answers: a) <span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /6, 2<span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /3 b) 5<span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /12, 11<span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π /12 c) 54°, 144° d) 67°, 157°

The second hand of a clock is 11.4 centimeters long. (Remember - a second hand takes 60 seconds to complete one revolution) How far does the tip of the hand travel in one minute? At what speed does the tip travel? What is the angular speed of the tip?

Answers: The tip travels 71.6 centimeters, or 22.8<span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π centimeters. It travels at the rate of 1.19 centimeters per second. The angular speed is <span style="FONT-SIZE: 12pt; FONT-FAMILY: 'Times New Roman'; mso-fareast-font-family: 'Times New Roman'; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA">π/30 radians per second, or 6° per second.