# Very basic trigonometry question?

## Homework Statement

Well basically i'm trying to learn trigonometry from a textbook. It shows a rule,
to change from degrees to radians, multiply by pi/180.
to change from radians to degrees, multiply by 180/pi .

## The Attempt at a Solution

I'm the type of person that wants to know why what works. I cannot sleep when i cannot understand the background work of an rule. why do those rules work? ty

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Dick
Homework Helper
It's because a full circle is 2*pi radians and it's also 360 degrees. So (2*pi radians)=(360 degrees). So (dividing both sides by 2*pi), 1 radian=(360/(2*pi)*degree=(180/pi)*degree. That's your first conversion, you do the second.

phinds
Gold Member
2019 Award
Dick's response is correct, but a more fundamental answer is that it is inherent in the definition of degrees and radians. They are just different units for the same thing, like miles and kilometers.

ehild
Homework Helper
The length of an arc s in a circle belonging to the central angle φ is s=(φ(degree)/360°) (2Rπ). Instead of degrees, we can measure the angle with the ratio of (arc length / radius): φ(radian)=s/R. Comparing with the previous equation φ(radian)=s/R=2π ( φ(degrees)/360°).

ehild