Question

How many degrees $=$ $1$ radian $?$

Answer 1

Hint : If we see any word like radius and degree in question, then to remember this consider a full circle around ourselves which is of complete $360^\circ$ and remember that $2\pi$ completes a full circle and $\pi$ have a value which is not in degrees. So, consider a simple statement that is $2\pi$ is equal to $360^\circ$ .By eliminating $2$ from both sides we get $\pi = 180^\circ$ .So, clearly one radian is equal to $180^\circ$ .In other words, to find the value of $1$ radian we have to know the value of $\pi$ radians. As $\pi$ radians are equal to $180^\circ$ . Therefore, we can say that $1$ radian is equal to $\dfrac{{180^\circ }}{\pi }$ .

Complete step-by-step answer :
Angle subtended at the centre by an arc of length $1$ unit in a unit circle $($ circle of radius $1$ unit $)$ is said to have a measure of $1$ radian.
$1$ revolution is equal to an angle of $2\pi$ radians.
i.e., $2\pi$ radians $=$ $360^\circ$ $=$ one revolution
and $\pi$ $=$ $180^\circ$
Hence, we can say that $180^\circ$ is equal to $\pi$ radians.
In the question, they asked the value of $1$ radian in degrees. So, let’s see how we can convert radians to degrees for any specific angle. To convert radians to degrees we use the formula:
$radians{\text{ }} \times {\text{ }}\dfrac{{\pi}}{180 }{\text{ }} = {\text{ }}\deg rees$
According to the question we have to convert $1$ radian to degrees.
$\therefore$ $1$ radian $=$ $1{\text{ }} \times {\text{ }}\dfrac{{180}}{{\dfrac{{22}}{7}}}$
$($value of $\pi$ $=$ $22/7$ i.e., $\approx$ $3.14$ $)$
$=$ $1{\text{ }} \times {\text{ }}\dfrac{{180{\text{ }} \times {\text{ 7}}}}{{22}}$
$=$ $\dfrac{{1260}}{{22}}$
Further simplifying we get
$=$ $57.2727273$
$=$ $\approx {\text{ }}57.3$
So, the correct answer is “$\approx {\text{ }}57.3$”.

Note : To convert radians to degrees: multiply by $180$ , divide by $\pi$ and to convert degrees to radians: multiply by $\pi$ , divide by $180$ . The radian is the fixed size no matter what the size of the circle is because the length of the arc is equal to the radius of the circle. If no units are listed for an angle measure, it is assumed to be in radians. Theta $\left( \theta \right)$ should be measured in radians. When working in the unit circle with radius $1$ , the length of the arc equals the radian measure of the angle.