Notes: 4.1: Angles and their Measures:
Trigonometry from Greek trigōnon "triangle" + metron "measure"
An angle is a ______________ of a _______________. In plane geometry, an angle is the measurement of
the __________________ required to move one ray, called the _____________________ onto another ray,
called the ______________________, emanating from the same point, called the __________________.
We typically use Greek letters to denote angles. We will use  , know as “theta” to denote both the
_____________ of angle as well as its _____________________________
Given an angle, we can draw an __________ that intersects the angle. We say the angle
_________________ the arc. The length of this arc is appropriately called the _______________________
and is denoted by the letter S (A was already taken.) When drawn as a wedge of a circle, the angle is called
the _________________________________.
Now that we know what an angle is, how to we measure it?
Method 1: In Degrees, Minutes, and Seconds (or DMS.)
We are all familiar with the fact that one full rotation equals 360 . From this, we can partition up any
rotation into fractions of 360 , and in fact, doing so in a counter-clockwise manner beginning on the
positive x-axis results in the division of the coordinate plane into 4 quadrants each of size 90 .
Example 1:
Draw an angle of 30
If we decide to rotate by fractions of full degrees, we can use either decimals or DMS notation.
If you’ve ever found a coordinate on a map or from a GPS device, you are already familiar with DMS
notation. It works much like keeping time on a clock in the sense that one full degree equals 60 minutes
(notated with a ‘ ) and 1 minute equals 60 seconds (notated with a “).
From this we can develop some conversion factors:
1
60
60
1
or
and
or
60
1
60
1
Example 2:
Draw an angle of 185.637 , then convert it to DMS.
You can do this exclusively on the calculator by utilizing the “Angle” menu, usually found as the second
operation on the “Apps” key or the “Matrix” key (83s and below).
Example 3:
Draw and angle of 145 1522 , then convert it to decimal degrees.
This can also be done exclusively on the calculator, but it requires a few more key stokes and the use of a
“hidden” key.
Example 4:
Draw an angle of 342 1412 , then convert it by hand to decimal degrees. Confirm with your calculator.
Historical note: the use of 360 and 60 minutes/seconds is not so arbitrary. It dates back to the Ancient Sumerians and
Babylonians who used a base 60 number system (called a Sexagecimal system). Why 60? We use base 10 likely because we
have 10 fingers. Did the Babylonians have 60 fingers?! Probably not. The number 60 has twelve factors, 1, 2, 3, 4, 5, 6, 10, 12,
15, 20, 30, 60, of which 2, 3, and 5 are prime. With so many factors, many fractions of sexagesimal numbers are simple.
Additionally, 60 is the smallest number divisible by every number from 1 to 6.
Using degrees is both fine and dandy. . . . sometimes, but it does have its drawbacks. It carries with it a
very specific unit of measure, the little “bubble” atop the angle measure. That means degrees themselves
are specific units of measure. There is, in fact, another way to measure rotation, a method that requires no
units. The need for such a way to measure rotation will become apparent by the end of this lesson.
Method 2: In Radians (or rads)
One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of
the circle.
Here’s an interesting question:
For any given circle with radius r, how many lengths of its radius can be wrapped around its
circumference?
In general, the radian measure of any angle is the ratio of its _______________________ to the
_________________ of the circle. We can write this mathematically as:
S

r
Do you see from this ratio why the radian is unitless????????????????????????????????????????
Here’s another interesting question:
What do we call the arc length if we go all the way around the circle?
The formula is__________________.
Using the formula for  above, we can arrive at the radian measure of a full rotation:
We now know that 360  1 rotation  2 . This gives us a way convert back and forth between the two.
180

So how many degrees is one radian anyway?
1 radian = _______________________ degrees
This is good to know!!
Let’s try to convert some more
or

180
Example 5:
Example 6:
Convert 60 to radians
Convert
Example 7:
Example 8:
Convert 47.632 to radians
Convert

to decimal degrees
4
4
to degrees
5
The equation for the radian measure of an angle, when altered slightly, gives us another useful formula that
is often misused but that also reveals the necessity of the radian’s existence as a unitless measure of angles.
S  r
Example 9:
Find the perimeter of a 60 slice of 14-inch pizza.
Sometimes we are not just interested with how much something has rotated, but also how FAST it rotates.
This is a measure of something’s Angular Velocity and can be measured in Revolutions Per Minute
(RPMs), Radians per unit of time, or Degrees per unit of time. The Tachometer in your car measures how
fast your crankshaft spins. On old (or new) record players, there was an RPM switch as different records
had to be played at different speeds (to sound correctly.)
Example 10: Super Stoichiometry
A fit, law-abiding math student rides his bike through Landa Park at the posted 20mph speed. If the tires
on the bike have a radius of 15 inches, how fast are his (or her) tires spinning in RPMs?