Chapter 12.2, Problem 53HP

To determine

Explain that one degree and one radian represents which part. Also explain the convert between the measures.

Answer to Problem 53HP

The one degree and one radian represent $1^{\circ }=\frac{1}{{180}^{\circ }}$ & $1radian=\frac{1}{\pi }$ .

Explanation of Solution

Given: It is given in the question that there is a circle whose radius is r.

Concept Used:

In this, use the concept that ${180}^{\circ }$ is measure in degree and $\pi$ is measure in radian.

Calculation:

Since, there is a circle with radius r.

Now, it represents $1^{\circ }=\frac{1}{{180}^{\circ }}$ of circle with radius r.

Also, it represents $1radian=\frac{1}{\pi }$ of a circle with radius r.

Since, by using the above fact . to convert between radians and degree use the general equation that ${180}^{\circ }=\pi$ .

Hence, that’s why to convert degree into radian ,multiplying by $\frac{{180}^{\circ }}{\pi }$ and when converting radian into degree, multiplying $\frac{\pi }{{180}^{\circ }}$ .

Conclusion:

The one degree and one radian are $1radian=\frac{1}{\pi }$ & $1^{\circ }=\frac{1}{{180}^{\circ }}$