The Trick To Converting Radians To Degrees

Let's look at a different scenario when $\pi$ is visible in the simplified expression. We've seen it used in the numerator, but what happens when $\pi$ in the denominator? Does it still work?

Let's see by first converting $\frac{1}{\pi}$ radians into degrees the normal way (Using dimensional analysis), and then convert using Jacob's trick. If the results are not equal, Jacob's trick would no longer hold.

Like I said, let's first convert using dimensional analysis.

$\frac{1 \;rad}{\pi}\times \frac{180^{\circ}}{\pi \;rad}$

Of course, the radian labels cancel out like common factors when multiplying fractions.

$\frac{1}{\pi}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{\pi^{2}}$

So, we know that $\frac{1}{\pi}$ radians is equal to $\frac{180}{\pi^{2}}$ degrees (Note that $\frac{180^{\circ}}{\pi^{2}}$ is equivalent to saying $(\frac{180}{\pi^{2}})^{\circ}$ ).

Let's now try using Jacob's trick.

$\frac{1}{\pi}\rightarrow \frac{1}{(180)}\rightarrow (\frac{1}{180})^{\circ}$

So, using Jacob's trick, our answer becomes $\frac{1}{180}$ degrees. Are the two answers equivalent?

Well, no. $\frac{180}{\pi^{2}}\approx 18.238$ and $\frac{1}{180}\approx 0.006$ . So, it is quite clear that $\frac{180}{\pi^{2}}^{\circ}\neq\frac{1}{180}^{\circ}$ and therefore Jacob's trick doesn't always work . In fact, this trick always works when $\pi$ is only on the numerator, but it doesn't usually work when $\pi$ is in the denominator.