Question

What‌ ‌is‌ ‌the‌ ‌general‌ ‌formula‌ ‌to‌ ‌convert‌ ‌radians‌ ‌to‌ ‌degrees‌ ‌and‌ ‌vice‌ ‌versa?‌ ‌

Answer 1

Hint: We will use the identity that relates the radian measurement and the degree measurement, ${{\pi }^{c}}=180{}^\circ .$ In order to convert radians to degrees, we need to multiply the value in radian with $\dfrac{180}{\pi }.$ If we want to convert degrees to radian, we need to multiply the value in degree with $\dfrac{\pi }{180}.$

Complete step-by-step solution:
We are asked to find the formula to convert radians to degrees and vice versa.
In both the cases, we will have to use the identity given by ${{\pi }^{c}}=180{}^\circ .$
Let us first discuss how to convert radians to degrees.
So, in that case, we need to find out the corresponding degree value to ${{1}^{c}}.$
We can easily find it using the given identity ${{\pi }^{c}}=180{}^\circ .$ We just need to transpose $\pi $ to the right-hand side. We will get ${{1}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}.$
So, we obtained the corresponding degree value to ${{1}^{c}}$ and from this, we can conclude that we need to multiply the value in radian measurement with ${{\left( \dfrac{180}{\pi } \right)}^{\circ }}.$
Now, we will discuss how to convert degrees to radians.
As we have already said, we will use the identity ${{\pi }^{c}}=180{}^\circ .$
Now, to find what $1{}^\circ $ in radians, we will transpose $180$ to the left-hand side.
We will get ${{\left( \dfrac{\pi }{180} \right)}^{c}}=1{}^\circ .$
Therefore, we will have to multiply the value in degree measurement with ${{\left( \dfrac{\pi }{180} \right)}^{\circ }}.$
Hence the formula for converting radians to degrees is ${{1}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$ and the formula for converting degrees to radians is ${{\left( \dfrac{\pi }{180} \right)}^{c}}=1{}^\circ .$

Note: We know that radians and degrees are measures of angles. We also know that the angle for an object to complete a full rotation is $360{}^\circ $ in degree measurement and $2{{\pi }^{c}}$ in radian measurement.