1. ## Reciprocating Radians & more

We have that

1 Radian = $\displaystyle \frac{180}{\pi}$ degrees

Then isn't it true that

$\displaystyle \frac{1}{1}=1$ Radians = $\displaystyle \frac{\pi}{180}$ degrees?

Also, I was reading in a precalc book that the area of a sector of a circle is $\displaystyle \frac{1}{2} \times r^2 \times \theta$, where $\displaystyle \theta$ is the angle in radians. As I'm not used to using radians, I usually do $\displaystyle \frac{r^2 \times \pi \times \Phi}{360}$ where $\displaystyle \Phi$ is the angle in degrees. Can someone please explain slowly how to transfer from one to the other?

2. Originally Posted by DivideBy0
We have that

1 Radian = $\displaystyle \frac{180}{\pi}$ degrees

Then isn't it true that

$\displaystyle \frac{1}{1}=1$ Radians = $\displaystyle \frac{\pi}{180}$ degrees?
So you are trying to say that
$\displaystyle 1\text{ rad } = \frac{1}{1 \text{ rad }}$??
If you have an angle of $\displaystyle \theta$ rad then it is equivalent to an angle of $\displaystyle \theta \cdot \frac{180^o}{\pi~\text{rad}}$ degrees.

If you have an angle of $\displaystyle \theta$ degrees then it is equivalent to an angle of $\displaystyle \theta \cdot \frac{\pi~\text{rad}}{180^o}$ rad.

Originally Posted by DivideBy0
Also, I was reading in a precalc book that the area of a sector of a circle is $\displaystyle \frac{1}{2} \times r^2 \times \theta$, where $\displaystyle \theta$ is the angle in radians. As I'm not used to using radians, I usually do $\displaystyle \frac{r^2 \times \pi \times \Phi}{360}$ where $\displaystyle \Phi$ is the angle in degrees. Can someone please explain slowly how to transfer from one to the other?
$\displaystyle A = \frac{1}{2} \times r^2 \times \theta$
where $\displaystyle \theta$ is in radians. You want to use an angle in degrees, so you must convert from $\displaystyle \Phi$ degrees to radians:
$\displaystyle A = \frac{1}{2} \times r^2 \times \left ( \Phi \cdot \frac{\pi}{180^o} \right )$

$\displaystyle A = \frac{1}{2} \times r^2 \times \left ( \Phi \cdot \frac{2\pi}{360^o} \right )$

and the 2 in the parenthesis cancels the (1/2) out front leaving
$\displaystyle A = r^2 \times \left ( \Phi \cdot \frac{\pi}{360^o} \right )$

-Dan

3. Thanks!