Measuring Angles in Degrees

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Pre-Calculus $\longrightarrow$
Trigonometry $\longrightarrow$
  • Specifically define / recall what exactly degrees are
  • Define partial degrees as fractions of degrees in a system called DMS (degrees minutes seconds)
  • Learn how to convert partial degrees between our natural decimal system and DMS measure
Lesson Description

At this point in your math career, you've probably measured angles using degrees in the past. This first lesson on angle measurement will reinforce what we already know about degrees, and also discuss a couple of common ways to work with partial degrees.

Practice Problems

Practice problems and worksheet coming soon!


Familiar Degrees

As early as you learned what an angle is, you learned that degrees is how we measure them. As we begin our study of trigonometry, we'll use degrees for what we've always used them for - measuring the size of angles - but we'll also be looking at angles in ways we probably haven't before.Let's formally define what degrees are, and remember some important common angle measures and their properties.
Define: DegreesOne degree is defined to be the central angle in the sector of a circle, for a sector that represents $1/360$-th of the circle.
A plainer way to say this is that there are $360^{\circ}$ in a circle - a fact that is probably not news to you.In grammar school as well as in a high school geometry class, we seem to repetitively work with certain "common" angles like $30^{\circ}$ and $45^{\circ}$. At this point, we will only need to understand that these angles tend to have nice properties because common multiples of those angles comprise a full circle. For example, if you slice a circle into eight pieces, you get sectors that each have a central angle of $45^{\circ}$. Similarly, slicing a circle into twelfths gives sectors with $30^{\circ}$ angles.You'll see many trigonometry topics focus on these angles for various reasons as you progress into advanced trig concepts. This fact is the foundation of many "nice" properties that these angles have.

Splitting Hairs

It's reasonable to say that a single degree is a small measure. However, the bigger the circle, the more you might care about a single degree of angle.For example, our global position standards use latitude and longitude to identify where you are on Earth. If you move one degree of longitude, you will have travelled about $69$ miles. On the other hand, if you and your friend each drew a perfect circle that filled up the whole page, and one of you drew in a perfect 34 degree sector and the other drew a perfect 33 degree sector, there's a good chance most of your classmates would have a hard time telling them apart.For situations where the difference of a degree is substantial, math has a standardized way to further subdivide degrees into smaller units. The bad news is that the system is not base $10$. The good news is that the system is based on something familiar - time.Although it is arguably a small measure on its own, one degree is further subdivided into $60$ equal pieces - and each of those small slices is further divided into another $60$ pieces.
Define: Arc-minutesOne arc-minute (sometimes called a degree minute) is an angle equal to $1/60$-th of a degree. It is notated with a single apostrophe.
For example, we could communicate $80.5^{\circ}$ as $80^{\circ}$ $30'$.
Define: Arc-secondsOne arc-second (sometimes called a degree second) is an angle equal to $1/60$-th of an arc-minute. It is notated with a double apostrophe or a standard quotation mark (two primes).
For example, we could communicate the angle of 42 degrees, 20 minutes, and 38 seconds as$$42^{\circ}\,20'\,38''$$There is no further granular unit of measure smaller than an arc-second. If you need to be more exact than an arc-second, we resort to decimals (e.g. $26^{\circ}$ $53'$ $31.52''$).
Vocab FYI
One arc-second is incredibly small - so much so that if you were standing on the top of the Prudential Center in Boston, and you could see a passenger car parked in Chicago, from your perspective the angle that sweeps through the width of that car's bumper is about one arc-second.On a global scale, if you move one arc-second in longitude, you've only moved about 100 feet.

Converting Decimals and DMS

As if we are all going to become GPS experts or military operatives, most curricula require us to know how to work with degrees, minutes, and seconds (DMS) to measure and understand angles. Part of this includes the ability to convert an angle in decimal form to an angle in DMS form. While this is a mildly annoying operation, it is not devastatingly difficult.Converting to DMSLet's say you start with an angle like $61.8392^{\circ}$. How do we get it into DMS form?The whole number of degrees is already known - $61$. In your calculator, you should subtract away the whole number of degrees, leaving only the decimal.To figure out how many minutes this represents, simply multiply by $60$.While we now know that $0.8392$ is $50.352$ arc-minutes, we want to turn the leftover decimal into arc-seconds. Similar to the prior step, we will subtract $50$ from our result, since we know we have $50$ whole arc-minutes and we want to know what number of arc-seconds the remaining decimal has.Once again multiply by $60$:Since arc-seconds is the smallest unit we'll use, we take this result as the final piece of our answer.$$61.8392^{\circ} = 61^{\circ} \, 50' \, 21.12''$$
If you are working with long decimals in your calculator, do not re-type or truncate them in for subsequent steps. Arc-seconds are very small, so any rounding can yield an incorrect answer. Use 2nd $\longrightarrow$ Ans to work with exact prior answers, or use your up and down arrows to select prior answers on screen.Do this:Not this:
Converting From DMSIf you have a DMS notated angle and you need to convert it back to a degree and decimal based measure, follow this rule:Degree decimal = whole degrees + [arc-minutes $\div$ $60$] + [arc-seconds $\div$ $3600$]A common reason you may need to do this is to make performing arithmetic on angles easier. It's much more manageable to add decimal form angles than it is to add DMS structure angles, because when you add DMS, you need to "carry over" units, just like with time. e.g. $40$ arc-seconds plus $30$ arc-seconds needs to be expressed as $1$ arc-minute, $10$ arc-seconds - not $70$ arc-seconds.

Put It To The Test

For each of the following exercises, convert the given degree measure notated in decimals to DMS measure.
Example 1$$92.75^{\circ}$$
Show solution
$\blacktriangleright$ Simply pick up your calculator and subtract $92$ from this expression, leaving you with $0.75$, and multiply by $60$.$$0.75 \cdot (60) = 45$$Since we have a whole number answer with no decimal extra / partial measure, we're actually done. $92.75^{\circ}$ is $92$ $45'$ in DMS measure.
Example 2$$114.732^{\circ}$$
Show solution
$\blacktriangleright$ Start by subtracting $114$ from this angle measure, and multiply the remaining $0.732$ by $60$:$$0.732 \cdot (60) = 43.92$$This tells us that $0.732$ degrees is $43$ whole arc-minutes plus a partial one. To get the partial arc-minute in terms of arc-seconds, let's subtract away the whole number and multiply what's left by $60$ once again.$$0.92 \cdot (60) = 55.2$$Our final answer is to say that $114.732^{\circ}$ can be expressed as $114^{\circ}$ $43'$ $55.2''$.
Example 3$$76.1\bar{6}^{\circ}$$
Show solution
$\blacktriangleright$ Do not give in to the temptation to punch in just a few $6$'s into your calculator. As we learned in Pre-Algebra, any decimal that repeats can be turned into a fraction, and if it's a relatively simple fraction, it may help us do less work!Either using your calculator or by any other method you know, we can find out that this fraction is really $1/6$. This is great news - if we want $76^{\circ}$ and one-sixth of a degree, then in DMS form, we need $1/6$ of $60$ arc-minutes, or exactly $10$ arc-minutes.Therefore, $76.1\bar{6}^{\circ}$ is equivalent to $76^{\circ}$ $10'$.
For the following angle measures in DMS measure, obtain the degree decimal form of the angle measurement.
Example 4$$120^{\circ} \, 33' \, 6''$$
Show solution
$\blacktriangleright$ To convert to decimal form, divide arc-minutes by $60$ and arc-seconds by $3600$.$$120 + \frac{33}{60} + \frac{6}{3600}$$$$=120.551\bar{6}$$So, $120^{\circ} \, 33' \, 6''$ is equivalent to $120.551\bar{6}^{\circ}$.
Example 5$$26^{\circ} \, 56' 41''$$
Show solution
$\blacktriangleright$ Once again, to convert DMS to pure decimal form, divide the arc-minutes by $60$ and the arc-seconds by $3600$.$$26 + \frac{56}{60} + {41}{3600}$$$$=26.9447\bar{2}$$Therefore, $26^{\circ} \, 56' 41''$ is equivalent to $26.9447\bar{2}^{\circ}$.
Lesson Takeaways
  • Retain what we already knew about using degrees to measure angles
  • Be able to conceptually describe what arc-minutes and arc-seconds are
  • Understand how important angle precision can be when the size of the application is very large, such as GPS location
  • Confidently convert back and forth from decimal notated partial angles to DMS form

Lesson Metrics

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