Video Transcript
An angular displacement of blank
degrees is equal to an angular displacement of 7.25 radians.
In this exercise, we want to fill
in that blank. In other words, we want to know how
many degrees 7.25 radians is. To convert this angle in radians
into some equivalent angle in degrees, we can recall that one complete revolution
around a circle is equal to two 𝜋 radians. And it’s also equal to 360
degrees. In this exercise, instead of having
two 𝜋 radians, our angle in radians is 7.25. So knowing that this is true, we
want to convert this value into degrees.
To see how to do that, we can
figure out what factor will go here so that the left-hand side of this equation is
equal to 360 degrees as we know it is. That is, we want two 𝜋 radians
times whatever it is that goes here to equal 360 degrees. We can see that if we put this
factor in parentheses, then the two 𝜋 radians will cancel out and we’ll be left on
both sides with 360 degrees. This tells us that we’ve discovered
the factor by which we can multiply an angular measure in radians to convert it to
degrees. It’s 360 degrees divided by two 𝜋
radians or, equivalently, 180 degrees divided by 𝜋 radians.
So that, then, is what we’ll
multiply 7.25 radians by. And when we do, notice that the
units of radians cancel out. And we’ll be left with an angle
measure in degrees. This is equal to 415.39 dot dot dot
degrees. But since our original angle is
given to us with three significant figures, we’ll keep that many in our answer. That is, we’ll keep this
significant figure, this one, and this one, which means our final answer is 415
degrees. So we can say that an angular
displacement of 415 degrees is equal to an angular displacement of 7.25 radians.
