Topic A: Lessons 8-10: Trigonometric graphs and radians
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We're asked to convert pi radians and negative pi/3 radians to degrees. And the first question I'll ask you: If you do one revolution, You have an angle that went all the way around once. How many radians is that? Well we know that it is 2 pi radians. Now that exact same angle if we were to measure it in degrees, How many degrees is that? Well if you were doing degrees, it would be one full revolution. That is equal to 360 degrees Now, can we simplify this? That's a bore to write this little, superscript circle That's literally the units of the question. Sometimes it doesn't look like a unit but it is a unit. You could literally write degrees instead of that little symbol. Now can we simplify this a little bit? Well sure, Both two pi and 360 are divisible by two so lets divide things by two, and if we do that, what do we get? Or what are pi radians equal to? Well on the left side here we're just left with pi radians, and on the righthand side here, 360 divided by two is 180. And we have still the units which are degrees. So we get pi radians are equal to 180 degrees. Which actually answers the first part of our question. We wanted to convert pi radians, well we just figured out! Pi radians are equal to 180 degrees. If you want to think about it, pi radians are halfway around the circle Halfway around the circle like that, and it is the same thing as 180 degrees. So now lets think about the second part. We want to convertnegative pi over three radians. --Switch to a new color-- so negative pi over three, so how do we convert that? So what do we get based on this information right over here. Well, to figure this out we need to know how many degrees there are per radian. We need to multiply this by degrees -- I'm going to write the word out instead of the circle here -- It would be really hard to visualize that, degrees per radian So how many degrees are there per radian? well we know that for 180 degrees we have pi radians. Or you can say there are 180 over pi degrees per radian. This is going to work out: We have however manyradians we have times the number of degrees per radian. So of course the units are going to work out. The radians cancel out, the pi also cancels. And you are left with 180 divided by 3, leaving us with what is that? Negative 60, and we don't want to forget the units We could write them out, the only unit left is degrees. WE could write out the word degrees or just put that symbol there.