Special right triangles radians how to#
And this allows us to figure out how to relate the entire degree system to the entire radian system. We can just write 2 pi and that implies that it is 2 pi radians. We don’t need to write r, or radians, or anything like that. Radians are so special that we don’t need a special symbol for them. So if 360 degree = 2 pi radians, notice that for degrees we have to use the degree symbol. 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. This means that there are 2 pi radians all the way around the circle. 2 pi radius lengths all the way around the circle. And so that means there are 2 pi radius lengths around the circumference.
We know that the circumference equals 2 pi r. In fact, we already measure all the way around the circle using radius lengths. 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. And we could use this as a unit of angle. This would define an angle of one radian. So all three sides of this little sector have the same length. And then the arc length also, even though it’s curved, it has the same length. And of course there are two radiuses going out from the center of the circle. Suppose we made an arc that had exactly the same length as the radius. So remember that arcs of circles have length. And this is precisely what we’re gonna do. 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. We can calculate everything we need from the radius. We can calculate the area from the radius, we can calculate the circumference from the radius. What is the most fundamental length associated with the circle, from which everything about the circle could be calculated? Obviously the radius. 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. These Babylonian values are also the reason we have 60 minutes in an hour and 60 seconds in a minute. 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. But we still hang on to these two numbers. It’s a very old religion that died out a long time ago. They believed all kinds of things, they believed a host of strange Gods, they sacrificed barnyard animals for various things. These ancient Babylonians believed that 60 and 360 were sacred numbers.
So here’s a picture of the Babylonian God, Marduk, and his pet dragon. 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. Now we can introduce the very important idea of radians.
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