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

 

[rwisz]

Hi there! It's great that you're trying to understand the background behind the rules in trigonometry. The reason why these rules work is because of the relationship between degrees and radians. Degrees and radians are two different ways of measuring angles. Degrees are based on dividing a circle into 360 equal parts, while radians are based on dividing a circle into 2π (or 360°) equal parts. So when converting from degrees to radians, you are essentially converting the angle measurement from a fraction of 360° to a fraction of 2π. This is why you multiply by π/180, which simplifies to 1/180. Similarly, when converting from radians to degrees, you are converting the angle measurement from a fraction of 2π to a fraction of 360°. This is why you multiply by 180/π, which simplifies to 180/3.14, or approximately 57.3. I hope this explanation helps you in your understanding of trigonometry. Keep up the good work!