to view answer.Radians and degrees are two measures of an angle. Thus, every angle has a measure in degrees and a different measure in radians – but they are just different units. It is important to be able to convert back and forth between radians and degrees, first because one may be more useful in a certain problem than the other, and second, so that you have a good familiarity with how they relate to one another.
What is a radian? One radian is the measure of an angle that is created by wrapping the radius of a circle along the outside of its circumference. Thus, since the circumference is equal to $$2 \cdot \pi \cdot radius$$, the distance around the outside of the entire circle is $$2\pi $$ radians. In degrees, we know that a circle can be measured as an angle of $$360^\circ $$, and thus, we have that $$2\pi \;radians\; = \;360^\circ $$.
Thus, $$\pi $$ radians equals one-half of a circle, or $$180^\circ $$. The number of degrees of an angle is proportional to the number of radians that measure that same angle. This proportion can be expressed as $${\Large\frac{{180^\circ }}{\pi }} = \frac{{{\text{Number of degrees}}}}{{{\text{Number of radians}}}}$$.
So, to convert radians to degrees, we just have to multiply radians by $${\Large\frac{{180^\circ }}{\pi }}$$:
degrees = radians $$ \cdot {\Large\frac{{180^\circ }}{\pi }}$$
To convert degrees to radians, we use the following form of the proportion:
radians = $$\frac{{{\text{degrees}}}}{\Large{{180}}} \cdot \pi $$
Which can also be written:
radians = degrees $$ \cdot {\Large\frac{\pi }{{180^\circ }}}$$
Convert the angle measure from radians to degrees:
$$m$$∠ $$A = 7\pi \;\;radians$$
Then, we just multiply this by $$180^\circ $$, since there are $$180^\circ $$ for every $$\pi $$ radians, and then we divide out the $$\pi $$ to get the degree measure:
degrees = $${\Large\frac{{7\pi \cdot 180^\circ }}{\pi }} = 7 \cdot 180^\circ = 1260^\circ $$
So, $$m$$ ∠ $$A = 1260^\circ $$
Convert the angle measure from radians to degrees:
$$m$$ ∠ $$B = 0.5\;\;radians$$
The number of radians does not have to be in terms of $$\pi $$. Remember that $$\pi $$ is just a number like any other real number. So, when we have a number of radians not in terms of $$\pi $$, we still perform the same conversion, because we still have a number of radians. This time it is 0.5. But again, we take the number of radians, multiply by $$180^\circ $$, and divide by $$\pi $$:
$${\Large\frac{{0.5 \cdot 180^\circ }}{\pi }} = \left( {{\Large\frac{{90}}{\pi }}} \right)^\circ \approx 28.65^\circ $$
So, 0.5 radians is approximately $$28.65^\circ $$. So: $$m$$ ∠ $$B \approx 28.65^\circ $$. We usually approximate the number of degrees, using an approximation for π.
Convert the angle measure from degrees to radians:
$$m$$∠ $$C = 210^\circ $$
So, this time, we want to find the number of half-circles that this degree measure includes. We do this by dividing it by $$180^\circ $$. Then, every half circle we have corresponds to $$\pi $$ radians, so we just multiply our result by $$\pi $$:
$$radians ={\Large \frac{{210^\circ }}{{180^\circ }}} \cdot \pi $$
At this point, we just need to simplify the fraction to have our exact answer. So:
$$radians = {\Large\frac{7}{6}}\pi $$
Thus, $$m$$∠ $$C ={\Large \frac{7}{6}}\pi \;radians$$.
We usually do not approximate the number of radians using the $$\pi $$ button on a calculator. Instead, leave your answer in terms of $$\pi $$, but simplify the fraction.
Some practice problems to check your skills:
(Problems 1-3) Convert the angle measure from radians to degrees:
1. $$m$$ ∠ $$D = 3\pi \;radians$$
Think about it for a moment and then access this link to view answer.
2. $$m$$∠ $$F ={\Large \frac{{5\pi }}{7}}\;radians$$
Think about it for a moment and then access this link to view answer.
3. $$m$$ ∠ $$ G = 17\;radians$$
Think about it for a moment and then access this link to view answer.
(Problems 4-6) Convert the angle measure from degrees to radians:
4. $$m$$ ∠ $$ H = 180^\circ $$
Think about it for a moment and then access this link to view answer.
5. $$m$$ ∠ $$ J = 120^\circ $$
Think about it for a moment and then access this link to view answer.
6. $$m$$ ∠ $$ K = 610^\circ $$
Think about it for a moment and then access this link to view answer.