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.

Notes for Homework 5, due Monday, Feb. 27

Degrees to DMS: Most scientific calculators has a function that will take a decimal number and convert it to degrees, minutes and seconds. Degrees and minutes will be whole numbers, but seconds might be a decimal. Not all the calculators round the seconds decimal to the same level, so I ask that it be rounded to the nearest tenth.

Example: if we take the inverse sine of 1/3, we get 19.47122063...°,which I ask to be rounded to four places after the decimal, so 19.4712°. If we take the unrounded answer and find the DMS conversion, we get 19° 28' 16.394" when using the TI-83 and 19° 28' 16.4" when using the TI-30XIIs. Because these calculators made by the same company don't round to the same level, I ask for the rounding to be to the nearest tenth of a second.

Note: Inverse sine and inverse tangent of negative numbers give negative angles in Quadrant IV. Add 360° to get a number between 0° and 360°.

Degrees to radians, both as a decimal number and a decimal time pi. At this point in the semester, your calculator should be DEGREE mode, so on any of the problems on the homework should start with taking the requested inverse trig function, converting the answer to a positive angle if necessary. Whether you have the angle in decimal degrees or DMS, you can now multiply the answer by pi/180 to get radians as a decimal number. In this case, you should get 0.339836909..., which I ask to be rounded to .3398. If you divide this by pi, you get .108173448..., which is the decimal number to be multiplied by pi. This means the four answers to the inverse sine of 1/3 are

Decimal degrees: 19.4712°
DMS: 19° 28' 16.4"
Radians: .3398
Radians as a multiple of pi: .1082pi

   

Practice for finding the other trig functions and the angle from the value of sine, cosine or tangent, assuming all are positive

Example #1: cosalpha = 2/5

(2/5)² + sin²alpha = 1
4/25 + sin²alpha = 1
sin²alpha = 21/25
sinalpha = sqrt(21)/5

since we have sine and cosine, tan = sin/cos.

tanalpha = [sqrt(21)/5]/[2/5] = sqrt(21)/2

With this editor, I can't write -1 as a superscript, so instead I will use the words arcsine, arccosine and arc tantangent. On your calculator, arccos(2/5) = 66.42182152...°, which we round to 66.4218°.

If you take the unrounded value and change it to degrees-minutes-seconds, rounding the seconds to the nearest whole number, we get 66° 25' 19".

Example #1.1: If we had sinalpha = 2/5, the work would look nearly identical, except cosalpha would equal sqrt(21)/5.  Tangent of this angle is the reciprocal of sqrt(21)/2, which is 2sqrt(21)/21 when written in rational denominator form.  The angle is the complement of our original angle, which means they add up to 90°. arcsin(2/5) = 23.57817848...°, which rounds to 23.5782°.

The DMS version of the unrounded value, rounded to the nearest whole second is 23° 34' 41".

Example #2: tanbeta = 2/5

Since tan = sin/cos,
tan × cos = sin

Using this, cos²beta + (2/5)²cos²beta = 1

cos²beta + 4/25cos²beta = 1
29/25cos²beta = 1
cos²beta =25/29
cosbeta = sqrt(25/29) = 5/sqrt(29) = 5sqrt(29)/29

arctan(2/5) = 21.80140949...°, which rounds to 21.8014°, and in DMS is 21° 48' 5".

Here are practice problems. Get the other two trig values, the angle rounded to the nearest ten thousandth of a degree and rounded to the nearest second in DMS mode.

1) cosalpha = 1/10

2) tanbeta = 1/10

3) singamma = 1/9

4) tandelta = 1/9

Answers in the comments.

Notes for Homework 5

Notes on decimal degrees, degrees/minutes/seconds (DMS), radians and radians written as multiples of pi

practice with fractions of a circle

Consider 17/19 of the circle as an angle, which we will call theta.

a) Write theta as an angle in decimal degrees, rounded to four places.
b) Write theta in DMS, rounded to nearest degree.
c) Write theta as an exact fraction of the circle using pi.
d) Write theta as radians written without pi, rounded to four places after the decimal.
e) sin theta
f) cos theta
g) tan theta
h) cot theta
i) sec theta
j) csc theta

Answers in the comments.

Notes on Homework 5, due Feb. 27

Notes on inverse trig functions

Notes on decimal degrees, degrees-minutes-seconds (DMS), radians and radians given as multiples of pi