Why Do We Convert Degrees to Radians in Trigonometry?

[freeofwork]

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 .

Homework Equations

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

 

[Dick]

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]

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]

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