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Video: Converting Degrees into Radians

Chris O’Reilly

Convert the following angle measures from degrees to radians. Give your answers in terms of 𝜋 in their simplest form. 90°, 30°, and 55°.

03:13

Video Transcript

Convert the following angle measures from degrees to radians. Give your answers in terms of 𝜋 in their simplest form. 90 degrees, 30 degrees, and 55 degrees.

And in order to actually convert the following angle measures, what we need to do is work out what our conversion factor is going to be. And to help us to do this, I’m just gonna help us understand how radians work. Well, we know that one full circle is equal to 360 degrees. But then if we look at this in radians, this is equal to two 𝜋 and then radians, which we denote with this little 𝑐. Okay, great. So we know that one full circle, 360 degrees, is equal to two 𝜋 radians. Okay, so what’s half a circle gonna be? Well, we know that half a circle or a semicircle is equal to 180 degrees. So therefore, this is gonna be equal to half our original amount of radians. So it’s gonna be equal to just 𝜋 radians. And it’s from this last relationship, where the 180 degrees is equal to 𝜋 radians, that we can actually work out our conversion factor. Because actually what we have is that the angle in radians is equal to the angle in degrees multiplied by 𝜋 over 180. Like we said, that’s we’ve got that from the previous relationship.

Okay, great. So now we know our conversion factor. Let’s get on. And let’s convert our angles. Okay, we’re gonna have a look at our first angle now. And we know that the angle in radians is gonna be equal to 90 degrees, which is our angle, multiplied by 𝜋 over 180. Well, this is gonna give us 90𝜋 over 180. And what, was this the final answer? Well, if we look at it, it is in terms of 𝜋, yes. But it’s not quite yet in its simplest form. So we’ve got one more step to go. And then we can have our final answer. So therefore, we can say that 90 degrees is equal to 𝜋 over two radians. Okay, great. Let’s move on to our next angle.

Okay, for our second angle, we can say that the angle in radians is equal to 30 degrees multiplied by 𝜋 over 180, which is gonna be equal to 30𝜋 over 180. And then now, if we fully simplify, we get that 30 degrees is equal to 𝜋 over six radians. Okay, great. Move on to our final measure.

So we get that the angle in radians is equal to 55 degrees. So that’s our angle multiplied by 𝜋 over 180, which will give us 55𝜋 over 180. Then again, we’re going to simplify because we want it in its simplest form. This one’s slightly trickier. But all we have to do for this one is divide the numerator and denominator both by five because it’s a factor of 55𝜋 and 180. And then we get our answer which is 11𝜋 over 36 radians.

Okay, great. So we can say that 90 degrees is equal to 𝜋 over two radians, 30 degrees is equal to 𝜋 over six radians, and 55 degrees is equal to 11𝜋 over 36 radians.