Radians to Degrees - Conversion, Formula, Examples
Radians and degrees conversion is a very crucial skill for progressive mathematics learners to understand.
First, we need to define what radians are so that you can understand how this formula works in practice. Then we’ll take it one step further by looking at a few examples of converting from radians to degrees easily !
What Is a Radian?
Radians are units of measurement for angles. It is derived from the Latin word "radix," which suggests ray or nostril, and is a fundamental idea in geometry and mathematics.
A radian is the SI (standard international) unit of measurement for angles, although a degree is a more frequently used unit in math.
In other words, radians and degrees are simply two distinct units of measure employed for measuring the identical thing: angles.
Note: a radian is not to be confused with a radius. They are two completely different things. A radius is the distance from the middle of a circle to the edge, while a radian is a measuring unit for angles.
Association Between Radian and Degrees
There are two ways to think regarding this question. The initial method is to contemplate about how many radians exists in a full circle. A full circle is equivalent to 360 degrees or two pi radians (exactly). Hence, we can state:
2π radians = 360 degrees
Or simply:
π radians = 180 degrees
The next way to think about this question is to calculate how many degrees exists in a radian. We all know that there are 360 degrees in a whole circle, and we also know that there are two pi radians in a full circle.
If we divide each side by π radians, we’ll notice that 1 radian is about 57.296 degrees.
π radiansπ radians = 180 degreesπ radians = 57.296 degrees
Both of these conversion factors are useful depending upon which you're trying to get.
How to Convert Radians to Degrees?
Now that we've covered what radians and degrees are, let's learn how to turn them!
The Formula for Converting Radians to Degrees
Proportions are a beneficial tool for turning a radian value into degrees.
π radiansx radians = 180 degreesy degrees
Simply put in your known values to obtain your unknown values. For example, if you wished to convert .7854 radians into degrees, your proportion would be:
π radians.7854 radians = 180 degreesz degrees
To find out the value of z, multiply 180 by .7854 and divide by 3.14 (pi): 45 degrees.
This formula works both ways. Let’s verify our work by reverting 45 degrees back to radians.
π radiansy radians = 180 degrees45 degrees
To solve for y, multiply 45 by 3.14 (pi) and divide by 180: .785 radians.
Since we've transformed one type, it will always work out with another straightforward calculation. In this instance, after converting .785 from its first form back again, following these steps created precisely what was predicted -45°.
The formulas work out like this:
Degrees = (180 * z radians) / π
Radians = (π * z degrees) / 180
Examples of Converting Radians to Degrees
Let's go through a few examples, so these theorems become simpler to digest.
At the moment, we will convert pi/12 rad into degrees. Much like previously, we will place this number in the radians slot of the formula and work it out it like this:
Degrees = (180 * (π/12)) / π
Now, let divide and multiply as you generally would:
Degrees = (180 * (π/12)) / π = 15 degrees.
There you have the result! pi/12 radians equals 15 degrees.
Let's try one more general conversion and transform 1.047 rad to degrees. One more time, use the formula to get started:
Degrees = (180 * 1.047) / π
One more time, you multiply and divide as suitable, and you will wind up with 60 degrees! (59.988 degrees to be almost exact).
Now, what to do if you have to convert degrees to radians?
By employing the very exact formula, you can do the converse in a pinch by solving for radians as the unknown.
For example, if you want to transform 60 degrees to radians, plug in the knowns and solve for the unknowns:
60 degrees = (180 * z radians) / π
(60 * π)/180 = 1.047 radians
If you remember the equation to solve for radians, you will get the same thing:
Radians = (π * z degrees) / 180
Radians = (π * 60 degrees) / 180
And there you have it! These are just a few examples of how to convert radians to degrees and vice versa. Bear in mind the equation and try it out for yourself the next time you need to make a conversion among radians and degrees.
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