SohCahToa and finding the angle of the Sun in the sky

In Friday’s post about the approach of the solstice I cheated and just looked up the Sun’s noontime altitude using Starry Night software. But the math geek in me decided to try to dredge up my high school trigonometry nearly 40 years since studying it. I’m happy to say that I’ve still got it!

SohCahToaI think that the only reason I remember SohCahToa is that we had a bunch of weisenheimers in our trig class, myself included, and we came up with a report called “SohCahToa East of Java,” a joke that required that you know your geological history and late-1960s disaster movies. SohCahToh is the standard mnemonic for remembering which values to use to figure sine, cosine, and tangent.

In our problem, we’re trying to figure the angle A, that of the Sun up in the sky, given that it’s above and behind a six-foot fence casting a 14-foot shadow. Since we know the length of the opposite side, the fence, and of the adjacent side, the shadow, SohCahToa says we use tangent: Tangent=opposite/adjacent. The inverse tangent gives you the angle, in this case 23.2 degrees. (Did you know that your iPhone has a trig calculator? I didn’t until I accidentally turned mine to landscape orientation when I was calculating something much simpler a few months back. So now you can figure triangle angles and trig functions wherever you go.)

Starry Night gave the Sun’s actual altitude that day as about 19.6 degrees. The difference can be attributed to the fact that the fence is a touch shorter than six feet, and the deck is raised off the ground a few inches, making the “opposite” side even a bit shorter. Close enough, though; I was out there in my jammies with a tape-measure, and it was about 20 degrees, so I was only taking approximate measurements!

But there you go. I even showed my work. SohCahToa!