ACT Math – Trigonometry Sample Question

Hipparchus, called “the father of trigonometry,” as imagined by Raphael in his famous fresco, “The School of Athens.”

One of the main differences between ACT Math and SAT Math is that the ACT tests trigonometry, while the SAT does not. So, what kinds of trig questions can you expect to see on the ACT? Consider the following typical problem:

Which of the following is equal to tan(Θ)cos(Θ) when sin(Θ) = (2/3) and 0<Θ<(π/2)?

There are two main ways you could go about solving this: with a calculator or without. Because I know how much you love your TI-83s, let’s try the calculator way first.

First, we need to find Θ, so we can type the following into the calculator:

sin-1(2/3) = .7297276562…

Store that monstrosity in your fancy calculator as letter A. Then just type in:

tan(A)cos(A) = .66666666667 = 2/3

Huh? It’s the same? Yes! If sin(Θ) = (2/3), then tan(Θ)cos(Θ) = (2/3), too. Isn’t it obvious? No? Well, let’s try the second method. In this method, we’ll be using your brain instead of your calculator.

It's all coming back to me now...
It’s all coming back to me now…

You may remember from math class some sort of crazy acronym like this: SOHCAHTOA. Or maybe you were supposed to memorize some weird sentence, like “Some Old Hippy Caught Another Hippy Trippin’ On Acid.”  Well, either way, what these mnemonic devices mean is:

sin(Θ) = opposite/hypotenuse

cos(Θ) = adjacent/hypotenuse

tan(Θ) = opposite/adjacent

If you had remembered that, then you would have been able to figure out that:

tan(Θ)cos(Θ) = (o/a)(a/h) = (o/h) = sin(Θ)

When you multiply tan(Θ)cos(Θ), the adjacents cancel out, leaving you with opposite/hypotenuse. Therefore:

tan(Θ)cos(Θ) = sin(Θ)

So of course they are the same!

"But I don't care if it's good for me!"
“But I don’t care if it’s good for me!”

This just goes to show that these questions are indeed written so that they can be done in under a minute WITHOUT a calculator, and indeed, sometimes it is faster to use your brain than it is to start punching in numbers. When you practice for the ACT, try working the problems without a calculator. Sure, you’ll have to go back and memorize your multiplication tables like you were supposed to in fifth grade, but I think you’ll find that doing the problems without the calculator, at least when you practice, will help you see the tricks that the test makers build these questions around, and will help you review the actual content more thoroughly. Besides, mental math is good for you!

Until next time, good luck and happy studying!

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