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Radian Measure of Angles - Problem 2
So you've learned the definition of radian measure. How many radians are in one revolution of a circle? Recall that the definition for radian measure as the theta equals s over r where s is the length of the intercepted arc and r is the radius of the circle.
In this circle I'm representing one revolution by theta, the radius of the circle is r and in this case the intercepted arc is the entire circumference, so I'll write this one down here s, and you remember that circumference is 2 pi r, so that's our arc length. Theta will then be 2 pi r over r and the r's cancel leaving 2 pi.
So there are 2 pi radians in one revolution and since there are also 360 degrees in one revolution, this gives us a way to convert from radians to degrees. 2 pi radians equals 360 degrees or you can divide both sides my 2 and use pi radians equals 180 degrees. Let's use this conversion in an example.
This problem asks us to convert to radians 30 degrees, 45 degree and 60 degrees. These three angles come up a lot in trigonometry so it's good to know their conversions to radians automatically. For 30 degrees just remember pi radians equals 180 degrees. We can divide 180 by 6 to get 30 so just divide both sides by 6 and that tells you that pi over 6 equals 30 degrees, and that's your answer pi over 6 radians.
Same thing for 45 degrees. Again start with pi equals 180 degrees and then just divide both sides by 4 because 180 divided by 4 is 45, if you can't remember that, remember that 180 divided by 2 is 90 and 90 divided by 2 is 45, so you divide both sides by 4 and get pi over 4 equals 45 degrees.
We'll use this trick one more time for 60 degrees, pi equals 180 divide both sides by 3, pi over 3 equals 60 degrees. So 30 degrees is the same as pi over 6 radians, 45 degrees is pi over 4, 60 degree is pi over 3, you'll see these angles a lot.