Degrees (including DMS) and radians (with or without multiples of pi.

We are now going to refer to the measure of angles in two different ways.  When we discuss angles of a triangle, for example, it is standard to give them in degrees and use the rule that the sum of the interior angles is 180°. It's very important to always have a degree sign on a number referring to degrees, because 30° is certainly not 30 meters or 30 feet or 30 of any measure we use for distance.

When we think of the trigonometric functions cosine and sine, we get numbers between -1 and 1 that correspond to the x and y values of a point on the unit circle. Instead of measuring an angle in degrees, when the trig functions are used in calculus and other settings, the angle is defined by arc length instead of degree.  The idea is that the distance around the unit circle (circumference) is equal to 2pi, where pi is the number your calculator represents as 3.141592654...  Here are some well known angles written in degrees and as multiples of pi and rounded to four places after the decimal.

360° = 2pi ~= 6.2832

270° = 3pi/2 ~=4.7124

180° = pi ~= 3.1416
120° = 2pi/3 ~= 2.0944
90° = pi/2 ~= 1.5708
60° = pi/3 ~=1.0472
45° = pi/4 ~=0.7854
30° = pi/6 ~= 0.5236

The formula to change from a° to radians is to multiply by the fraction pi/180.


One way to think of this is if you were to walk about 6.2832 meters around a circle with a 1 meter radius, you would come back to your original starting place.  This means 1 radian is a little bit less than 1/6 the way around the circle.  Let's turn "nice" radian numbers into degrees to the nearest thousandth. (The formula is to multiply by the fraction 180/pi.)

1 radian ~= 57.2958°
2 radians ~= 114.5916°
3 radians ~= 171.8873°
4 radians ~= 229.1831°
5 radians ~= 286.4789°
6 radians ~= 343.7747°
7 radians ~= 401.0705° or 41.0705°

At 7 radians, we have traveled more than 360°, so we can subtract 360° to get an angle that is easier to read.


If I write a decimal degree rounded to four places after the decimal, this is to the nearest ten thousandth, which is very small slice of an entire circle.  Another way to write degrees that are not whole numbers is degrees-minutes-seconds or DMS.  A minute is 1/60 of a degree and a second is 1/60 or a minute. (1/60)(1/60) is 1/3600, so if we write an angle using this method, it is not quite as precise as rounding to four places after the decimal (nearest 1/10,000), but more precise than three places after the decimal (nearest 1/1,000).

In class, I showed a way to do these by hand, but I missed that the calculator can do this for us.  There is a button (third button, second row from the top) with the symbols ° ' ".  If I want to change 57.2958° to DMS, I can type in 57.2958, press the [° ' "] button and scroll all the way to the right to find |>DMS instruction.  When I press [ENTER] the answer line says

57° 17' 44.9"

Because this rounds to the nearest tenth of a second, this is slightly more precise than four places after the decimal, but not as precise as five.

Also, if an angle is given in DMS, we can change back to decimal degrees by using the symbols °, ' and ".  This way I can type in 57° 17' 44.9", press [ENTER] and get 57.2958° back.

If I type in the formula to change a single radian to a degree I type 180/pi = 47.29577951...; asking for DMS of this gives us

57° 17' 44.8"

So we can see there was some rounding error at four places after the decimal.  If this is typed in and [ENTER] is pressed, we get

57° 17' 44.8" ~= 57.29577777..., which is not exactly the number we typed in.

Problems

Write these fractions of the circle as

decimal degrees (rounded to four places after the decimal)
DMS (rounded to nearest second)
radians as a multiple of pi (rounded to four places after the decimal)
radians (rounded to four places after the decimal)


a) 11/16 of the circle
b) 13/25 of the circle
c) 7/50 of the circle


Answers in the comments.