Convert the following angle
measures from radians to degrees. Give your answers correct to the
nearest degree where necessary. 𝜋 over three radians, 𝜋 over
seven radians, and three 𝜋 over eight radians.
Let’s recall what we already know
about radians. One full turn is equal to both 360
degrees and two 𝜋 radians. Halving this gives us 𝜋 radians
equals 180 degrees. Let’s take this relationship
further. Dividing both sides by 𝜋 will give
us the value of one radian. 180 over 𝜋 degrees is therefore
equal to one radian. So to change radians into degrees,
we can simply multiply by 180 over 𝜋.
Let’s see how this helps us. Multiplying by 180 over 𝜋 in our
first example gives us 𝜋 over three times 180 over 𝜋. Did you notice that there’s a
common factor of 𝜋? We can therefore cancel this
through, leaving us with a third times 180 over one. Multiplying as normal gives us 180
over three or 60 degrees. 𝜋 over three radians is therefore
equal to 60 degrees.
Now, consider our second
example. Once again, we can multiply by 180
over 𝜋 to convert our radians into degrees. Just like in the last example, we
have a common factor of 𝜋. We can therefore divide through by
𝜋 to leave us with a seventh times 180 over one. Multiplying here gives us 180 over
seven, which doesn’t give us such a nice number. However, the question told us to
give our answers correct to the nearest degree. 180 divided by seven is 25.714 and
so on. This gives us 26 degrees correct to
the nearest degree. 𝜋 over seven radians is therefore
equal to 26 degrees correct to the nearest degree.
At this point, you may wish to
pause the video and give the final example a go yourself. Just like in the two previous
examples, we are gonna start by multiplying by 180 over 𝜋. Once again, we can divide through
by this common factor of 𝜋. This leaves us with three-eighths
times 180 over one. Three-eighths times 180 over one is
equal to 67.5. Correct to the nearest degree, this
is 68 degrees. Three 𝜋 over eight radians is
therefore equal to 68 degrees correct to the nearest degree.