some example radians problems!

Important note here: please DON’T just try to speedrun these. Please don’t mechanically go through them as fast as possible—e.g., don’t try to blaze through all 44 problems, all at once! Yes, I know that the radians to-and-from degrees conversion is mechanical and formulaic, but I want us to do a bunch of conversions, spread out over time, not all at once—that’s how we learn the best! Not by doing six hours of soccer drills once a week—but by doing an hour of drills once a day. Or something like that. In other words, we want more repetition, over time! So even though these problems are “easy,” please don’t just burn through them as quick as possible! Our goal is to be able to convert between radians and degrees in our heads, totally fluently and fluidly and intuitively. We want to feel this stuff! And for that we need lots of practice!

\(2\pi\)

\(2\pi\)! That’s a whole circle! \(360^\circ\)!!! Yay! (Note here how I’m NOT just plugging things into the formula—I could, but I’m not—I’m trying to really FEEL how many degrees \(2\pi\) radians is!)

\[\boxed{ 2\pi \text{ radians } = 360^\circ \text{ degrees} }\]

\(11\pi/6\)

\(11\pi/6\)! That’s like ALMOST \(12\pi/6\), which would be \(2\pi\), which would be \(360^\circ\)… but it’s minus \(1\pi/6\). Meanwhile, \(1\pi/6\) is just \(\pi/6\) which is just \(30^\circ\), so this is \(360^\circ - 30^\circ = 330^\circ\).

\[\begin{align*} \frac{11\pi}{6} &= \frac{12\pi}{6} - \frac{1\pi}{6} \\ \\ &= 2\pi - \frac{\pi}{6} \\ \\ &= 360^\circ - 30^\circ \\ &= 330^\circ \end{align*}\]

\[\boxed{ 11\pi/6 \text{ radians } = 330^\circ \text{ degrees} }\]

\(\pi/3\)

Okay, we know that \(\pi\) radians is half of \(2\pi\) radians, which is \(360^\circ\), so \(\pi\) radians is \(180^\circ\). And \(180/3=60\). So this is \(60^\circ\)!

\[\boxed{ \pi/3 \text{ radians } = 60^\circ \text{ degrees} }\]

\(5\pi/2\)

OK, this is basically \(\pi/2\), i.e. \(90^\circ\), but plus another full revolution/full circle/\(2\pi\)/\(360^\circ\): \[\begin{align*} \frac{5\pi}{2} \text{ radians} &= \frac{4\pi}{2} + \frac{\pi}{2} \text{ radians} \\ \\ &= 2\pi + \frac{\pi}{2} \text{ radians} \\ \\ &= 360^\circ + 90^\circ \\ &= 450^\circ \end{align*}\]

\[\boxed{ 5\pi/2 \text{ radians } = 450^\circ \text{ degrees} }\]

\(60^\circ\)

Oooh, we already figured this out (in the reverse direction)! It’s \(\pi/3\).

\[\boxed{ \pi/3 \text{ radians } = 60^\circ \text{ degrees} }\]

\(135^\circ\)

Hmm, what’s this? We could use the formula… but we want to really FEEL this, so I’ll tell you how I would think about it. \(135^\circ\) is \(90^\circ + 45^\circ\). And we know what each of those angles are individually! \(90^\circ\) is \(\pi/2\) radians; \(45^\circ\) is \(\pi/4\) radians. So this is: \[\begin{align*} 135^\circ &= 90^\circ + 45^\circ \\ \\ &= \frac{\pi}{2} + \frac{\pi}{4} \\ \\ &= \frac{2\pi}{4} + \frac{\pi}{4} \quad\text{(fractions)} \\ \\ &= \frac{3\pi}{4} \end{align*}\] So we have:

\[\boxed{ 135^\circ \text{ degrees} = \frac{3\pi}{4} \text{ radians} }\]

\(179^\circ\)

Hahahahaha okay, this is a gross number of degrees. It’s ALMOST \(2\pi\) radians… but just a SMIDGEN less. I’ll use dimensional analysis to work it out: \[\begin{align*} &\left(\frac{179^\circ \text{ degrees} }{1}\right)\left(\frac{2\pi \text{ radians }}{360^\circ \text{ degrees}} \right) \\ \\ =&\left(\frac{179^\circ \cancel{\text{ degrees}} }{1}\right)\left(\frac{2\pi \text{ radians }}{360^\circ\cancel{\text{ degrees}}} \right) \\ \\ =& \frac{179\cdot 2\pi}{360} \text{ radians} \\ \\ =& \frac{358\pi}{360} \text{ radians} \\ \\ =& \frac{179}{180} \text{ radians} \\ \\ =& .99\overline{4} \pi \text{ radians} \\\\ \approx& 3.1241 \text{ radians} \end{align*}\] This should make sense—\(180^\circ\) is \(\pi\) radians; \(179^\circ\) is a little bit less than \(180^\circ\); and \(3.1241\) is a little bit less than \(\pi\) (a little bit less than \(3.1415\)).

\[\boxed{ 179^\circ \text{ degrees} = \frac{179}{180} \text{ radians} \approx 3.1241 \text{ radians} }\]