How to transform from degrees to radians and from radians to degrees

When dealing with trigonometric functions we often have to switch between the two units of measurement for angles: degrees and radians.

The degree representation of an angle is more convenient to use because we can imagine how a 90° or a 30° angle looks like.

Right triangle with angle notation

Image: Right triangle with angle notation

Even if we use degrees for trigonometry calculation, most of the software programs which can compute trigonometric functions expect the argument of the angle in radians.

For example in order to calculate the sine and cosine for the 90° angle in C we use the following instructions:

#include <stdio.h>
#include <math.h>
#define PI (3.141592653589793)

int main(int argc, char **argv)
 printf("Sine of 90 degrees is %.1f\n",sin(PI/2));
 printf("Cosine of 90 degrees is: %.1f\n",cos(PI/2));
 return 0;

In order to transform from degrees in radians we only need to know how much 180° or 360° is in radians. If we look at the unitary circle below we can see that 180° = π and 360° = 2π.

Trigonometric circle with four quadrants

Image: Trigonometric circle with four quadrants

To convert from degrees to radians we have to multiply the degree value with π and divide with 180.

\[ \begin{equation} \begin{split}
\bbox[#FFFF9D]{\text{radians}=\text{degrees} \cdot \frac{\pi}{180}}
\end{split} \end{equation} \]

Example 1: Convert 60° in radians:

\[60 \cdot \frac{3.1416}{180} = 1.0472\]

This means that 60° is equivalent with 1.0472 radians.

To convert from radians to degrees we have to multiply the radian value with 180 and divide with π.

\[ \begin{equation} \begin{split}
\bbox[#FFFF9D]{\text{degrees}=\text{radians} \cdot \frac{180}{\pi}}
\end{split} \end{equation} \]

Example 2: Convert 1.0472 radians in degrees:

\[1.0472 \cdot \frac{180}{3.1416} = 60\]

This means that 1.0472 radians is equivalent with 60°.

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