Radian Measure and the Unit Circle

[Sean Cook]

I need some guidance into understanding Radian Measure and the Unit Circle. This was the topic where I tanked and had to drop the course. I'm going to pick it up again next fall and want to start preparing now.

Any help is appreciated.

Sean

 

[HallsofIvy]

What, exactly, do you want to know? "Radian Measure and the Unit Circle" is a wide topic!

Essentially the idea of radian measure is that it IS the circumference around a unit circle cut off by an angle at the center. In particular, since a circle of radius 1 has total circumference of $2\pi$, radian measure in a circle goes from 0 to $2\pi$. A right angle cuts off 1/4 of the circle and so its measure is $2\pi/4= \pi/2$ radians. A "straight angle" cuts off 1/2 the circle and so its measure is $2\pi/2= \pi$ radians.

 

[Sean Cook]

That makes a lot more sense to me than the way is was explained, but it leads to my next trouble spot. How is a radian measured in degrees? This was where I was told to think of it like a "Clock" and that confused me and was unable to perform the calculations.

Sean

 

[HallsofIvy]

A complete circle is $2\pi$ radians or 360 degrees. You can think of that as "$2\pi$ radians per degree" or
[tex]\frac{2\pi \text{radians}}{360 \text{degrees}}[/itex] So to go from degrees to radians you multiply by $2\pi/360$ degrees to "cancel" the degrees and get radians. That is 90 degrees is $(2\pi/360)(90)= \pi/2$ radians.<br /> <br /> Going the other way, you just invert the fraction:<br /> [tex]\frac{360}{2\pi}[/itex].<br /> <br /> $\pi/3$ radians corresponds to $360/(2\pi)(\pi/3)= 360/6= 60$ degrees.[/tex][/tex]

 

[Sean Cook]

Thank you for explaining it this way, with the examples as well. I think I actually have a better understanding now :-)