3.2 Applications of Radian Measure

Before we can get into the applications of radian measure, first we must define what exactly a radian is.

In simple terms, a radian is a unit of measurement, similar to degrees but relating primarily to circles; however, we do use radians in trigonometric functions.

A radian is an angle whose corresponding arc in a circle is equal to the radius of the circle.

A radian measurement of a circle is directly proportional to the radius and the arc length.

It is different from the radius, which is half the diameter of the circle (the line that runs through the middle of a circle).

Say we have a circle with a radius of 1, the arc length would be 1 unit for an angle of 1 radian.

Say we have a circle with a radius of 5, the arc length would be 5 units for an angle of 1 radian.

Radians can be conceptualized in the same way that degrees can be, they’re just different!

If all of this seems confusing, and you would like to learn more, I highly encourage you to watch this video. It is 18 minutes long, but just the first 4 minutes are enough to help you start to conceptualize the meaning of what a radian is.

Moving forward.

Formula for Arc Length

\text{s = r}\theta

When you are working this equation, it is important to know that your s is equal to the arc length, and r is equal to the radius.

Remember that theta is in radians! When you are working these problems on your calculator, it must be in radian mode.

Formula for Area of a Sector

\text{A = }\frac{1}{2}\text{r}^2\theta

Again, remember to stay in radians.

In this equation, A is equal to the area, and again, r is equal to the radius.

Examples

1.

Find s.

\text{r = 8, } \theta = \frac{5\pi}{4}rad

Solution.

Use the arc length formula. Plug in known variables and solve for s.

\text{s = r }\theta = \frac{8}{1}(\frac{5\pi}{4}rad)

= 2(5\pi) = 10\pi

2.

Find theta.

\text{s = 6, r = 4, }\theta = ?

Solution.

\text{s = r}\theta \\ \frac{\text{s}}{\text{r}} = \frac{\text{r}\theta}{\text{r}}

\theta = \frac{\text{s}}{\text{r}} = \frac{6}{4} = \frac {3}{2} = 1.5\text{rad}

3.

Find A.

\text{A = ?, r = 7, s = 4}\pi

Solution.

Use the formula for finding the arc length to solve for theta. The r’s will cancel each other out and just leave theta and s/r.

\text{s = r}\theta

\frac{\text{s}}{\text{r}} = \frac{\text{r}\theta}{\text{r}}

\theta = \frac{\text{s}}{\text{r}} = \frac{4\pi}{7}\text{rad}

Now that we have the value for theta, we can plug in all the values for the sector area formula.

\text{A = } \frac{1}{2}\text{r}^2\theta

= \frac{1}{2}(7^2)(\frac{4\pi}{7})

The one of the squared sevens can cancel out with the other 7, and we are left with:

\frac{1}{2}(7)(4\pi) = 14\pi