We’ll continue our lessons on Trigonometry by looking at degrees and radians.

Now we can introduce the very important idea of radians. And I’ll start by saying, have you ever wondered why a right angle is defined as 90 degrees? Why that number 90? Where did the idea of degrees come from? The system goes back to the ancient Babylonian religion, over 3000 years ago.

So here’s a picture of the Babylonian God, Marduk, and his pet dragon. These ancient Babylonians believed that 60 and 360 were sacred numbers. They believed all kinds of things, they believed a host of strange Gods, they sacrificed barnyard animals for various things. It’s a very old religion that died out a long time ago. But we still hang on to these two numbers.

Turns out that the Babylonians thought all-the-way around a circle should divided into 360 parts, so that’s the origin of the degree system. These Babylonian values are also the reason we have 60 minutes in an hour and 60 seconds in a minute. So it turns out that even though this religion has been dead for millennium, turns out their values are still influencing a lot about how we look at the world.

Restored Ancient Babylon
By Jukka Palm

The fact that Babylonians decided to measure angles this way, based on the values of their ancient pagan religion, that does not necessarily mean that, mathematically, it’s the best way to measure angles. Degrees are familiar and commonly used, but they are not mathematically the best way to measure angles. So suppose we started from scratch, and we wanted to come up with the best way to measure angles.

The best way to measure angles, the most natural way to measure angles, derives not from some arbitrary choice based on a pagan religion. But it would derive from qualities of the circle itself. That’s the way the mathematician thinks, what are the qualities of the circle itself? Think about a circle.

Image By kabzarchyk

What is the most fundamental length associated with the circle, from which everything about the circle could be calculated? Obviously the radius. We can calculate the area from the radius, we can calculate the circumference from the radius. We can calculate everything we need from the radius.

If we could use the radius of a circle to define an angular measurement, this could be the basis of a new system for measuring angles. And this is precisely what we’re gonna do. This system is known as radians. So remember that arcs of circles have length. Suppose we made an arc that had exactly the same length as the radius.

So here, we have a little sector. And of course there are two radiuses going out from the center of the circle. And then the arc length also, even though it’s curved, it has the same length. So all three sides of this little sector have the same length. This would define an angle of one radian. And we could use this as a unit of angle.

So we could measure radians around the circle, and also that would be the same thing as measuring the number of radiuses around the circle. In fact, we already measure all the way around the circle using radius lengths. We know that the circumference equals 2 pi r. And so that means there are 2 pi radius lengths around the circumference. 2 pi radius lengths all the way around the circle.

This means that there are 2 pi radians all the way around the circle. So all the way around the circle, the thing that we used to call 360 degrees, we’re now gonna call that 2 pi radians. So if 360 degree = 2 pi radians, notice that for degrees we have to use the degree symbol. Radians are so special that we don’t need a special symbol for them.

We don’t need to write r, or radians, or anything like that. We can just write 2 pi and that implies that it is 2 pi radians. And this allows us to figure out how to relate the entire degree system to the entire radian system. So divide that equation by 2, we get 180 = pi, that’s very important. 180 degree angle is the same as a pi radian angle.

Divide that again, a right angle which we use to know as 90 degrees, we can now call that an angle of pi/2. So that is the measure over right angle in radians, pi/2. Now start with that right angle, if we divide that by 2, then we get a 45 degree angle and that’s pi/4. Again go back to the pi/2, instead divide it by 3.

We get 30 degrees = pi/6, if we multiply both sides by 2, we get 60 degrees = pi/3. These are the values of all the special triangles angles in radians. So the two special triangles one of them is a pi/4, pi/4, pi/2 triangle. The other is a pi/6, pi/3, pi/2 triangle. The equation, 180 degrees = pi is often a good place to start because 180 has so many factors.

Suppose we needed to find, say, 20 degrees in radians. Well of course, 180 = 20 x 9, and of course that equals pi. Just divide both sides by 9 and we get 20 degrees = pi/9. This can be a very quick way if the number of degrees is an obvious factor of 180. We can also change the equation 180 degrees = pi into a fraction equal to 1.

So 1 = pi/180, we can multiply any degree measurement by this fraction to change from degrees to radians. For example, 270 degrees multiplied by pi/180, we get 270 pi/180, which is 27/18. Divide top and bottom by 9, we get 3 pi/2, so 3 pi/2 is 270 degrees.

In fact, we can define the four quadrants in terms or radians. The first quadrant is 0 to pi /2, 0 to a right angle. The second quadrant is pi/2 to pi. A right angle to a full semicricle. Third quadrant is pi to 3pi/2. The first quadrant is 3pi/2 to 2pi.

## Identifying

This is often how the test will specify a particular quadrant. They won’t say third quadrant, they will say theta is between pi and 3pi/2. Which of the following is the value of 240 degrees in radians? Pause the video and then we’ll talk about this.

### Method One

Okay, so method one, this is a more visual method. Notice that 240 degrees is twice of 120 degrees, and 120 degrees is a third of the circle.

This means that 240 degrees is 2/3 of the circle. A whole circle is 2 pi, so 2/3 of 2 pi would be 4/3 pi. So that’s what 240 degrees is in radians. That’s method one.

### Method Two

Method two would be the more methodical method to take that number and multiply by pi/180.

When we multiply, we cancel the 0’s, then we cancel a factor of 6, and we get 4 pi/3. And so it turns out that the answer is answer choice D. Let me just point out, it’s very methodical to just multiply by pi/180. I’m going to discourage you from making your 100% of the time go to method for converting to radians.

It’s much more efficient to think in terms of proportions, and to think visually about the problem. You will understand radians much more deeply if you’re not simply reliant on multiplying by that ratio every time. The radian values of pi are easy to relate to a visual understanding. A 3pi/4 angle is 3/4 of pi, that means it’s 3/4 of a semicircle.

So it’s very easy to imagine. Take a semicircle, divide it into quarters, 3 of those quarters is a 3 pi/4 angle. To understand radians deeply when you can think about them visually as well as numerically. The full advantages of radians are not necessarily obvious at this level.

They become far more evident in calculus, when we start doing calculus with trigonometric functions. That’s where radians really prove their full value. Obviously, that puts us well beyond anything on the ACT, so we’re not gonna have to worry about that for the ACT. That’s just a preview for why radians are so important.

## Summary

In summary, degrees are the common way to measure angles, but radians are a mathematically preferable way to do so. The number of radians in the central angle of the circle equals the number of radius-lengths in its arc. This means that 360 degrees all the way around the circle is 2 pi radians and a 180 degrees, a semicircle, is pi radians.

To change from degrees to radians, we can think about simple proportions, or we can just multiply the number of degrees by the fraction pi/180.

## Author

• Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as "member of the month" for over two years at GMAT Club. Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test.