IMHO, I'd love for some math whiz to simplify this trig equation...

In My Humble Opinion
1

I googled for “cosine cubed law” and several variants and receive very few hits. That always makes me suspicious.

I’m looking at an explanation of why parking lot lights can be lower and require fewer fixtures. (There’s a thread in GQ if you’re really interested; but, it is not exciting.) Anyway, the explanation (www.darksky.org/infoshts/is078.html) has this cosine cubed law. If Þ is an angle, it’s saying that the equation

E = I × cos Þ ÷ ( H ÷ cos Þ )[sup]2[/sup]

simplifies to

E = I × cos[sup]3[/sup] Þ ÷ H[sup]2[/sup].

Now, I am more curious than suspicious, I guess. It is a specialized question, after all. I’d just really like to see how it’s done. Can anyone take me through the simplification? I never took trig and had to wing all the trig stuff in my later classes—it was a very seat-of-the-pants learning experience.

Thanks much.

2

There’s no trigonometry involved in this one. Just use the facts that (a/b)[sup]2[/sup] = a[sup]2[/sup]/b[sup]2[/sup], and that a/(b/c) = ac/b.

3

It’s things like this that keep us, i.e. me, humble.

I don’t recall the economist’s name, but he was a giant in the field. He published a paper in which he goofed on his second-order conditions and instead of finding an optimum he found a pessimum. This is what sets him apart. When the journal published a letter pointing this out, he published a reply that turned the whole thing into a learning exercise about keeping track of fundamentals.

Probably apocryphal, I suppose, but still a good story. Not that I’m trying to do the same thing. I outright goofed.