Ah, ACT Trigonometry. I can hear your reactions from here, my lovely Magooshers. “Oh, wow, trigonometry is one of the ACT math topics tested? Let me do my best cheerleader cheer! Cosine, secant, tangent, sine, 3.14159! Goooo Trig! Woo-hoo!”
Okay, I know most of you aren’t reacting that way (…yet), but I promise you’ll quake in fear just a little less when this is all over.
Note: For those of you who want another take on ACT Trig, look no further than this link.
And for more helpful math formulas, including ones for trigonometry, look here
To reiterate what we’re all going on about, we’ll need to review the basics of trigonometry, as far as the ACT is concerned. ACT Trig is pretty much concerned with right triangles and little else. If you like right triangles, you’re going to do well here.
To help illustrate my next point, let me tell you a brief story.
Once upon a time, there was a young man. He wanted to practice his baseball skills, so he started with throwing and catching. First he threw golf balls high into the air to see how high he could throw them without missing a catch. After a while, he got quite good at throwing golf balls, so he moved on to tennis balls. Once he felt confident enough with tennis balls, he moved to actual baseballs. Again, he became quite skilled at throwing baseballs, and decided to practice with bowling balls to keep improving his arm.
Of course, throwing bowling balls straight up into the air is not, generally speaking, standard practice for an aspiring baseball player, and he dropped the bowling ball directly onto the big toe of his right foot. He went to the hospital and met a lovely German doctor who told him that, luckily, his toe wasn’t broken, but he would have to take care of himself until he healed completely. He asked the doctor what he should do to take care of his foot. The doctor replied, “You must SOHCAHTOA.”
I know, I know, that was terrible. I hang my head in shame for the awfulness of that joke. But seriously, SOHCAHTOA is the answer to your trigonometry fears. It is an acronym that tells you everything you need to know to figure out basic trigonometry problems. It means:
Sine = Opposite / Hypotenuse (SOH)
Cosine = Adjacent / Hypotenuse (CAH)
Tangent = Opposite / Adjacent (TOA)
So, if you were looking for the cosine of a particular angle, you would take the value of the adjacent side to the angle and divide it into the value of the hypotenuse. Remember to keep things from the right point of view. Opposite always means “opposite to the angle you’re being asked about” and adjacent always means “next to the angle you’re being asked about.”
You might also have to deal with reciprocal trig identities. Again, the above link details this very well, but for ease of reading, I’ll explain here, too. Basically, these are the flipped-around versions of SOHCAHTOA.
Cosecant = Hypotenuse / Opposite
Secant = Hypotenuse / Adjacent
Cotangent = Adjacent / Opposite
I don’t know why sine and cosecant are reciprocals (or why cosine and secant are paired up), but I’ve found it’s always best not to pry too deeply into other people’s relationships.
To illustrate, take a look at this lovely little triangle:
So if we were looking for it would be . Make sense? Let’s list out our SOHCAHTOA, to make it easier to see.
“But wait!” I hear you cry, “What about C?”
I’m glad you asked. C, as we already know, is a right angle. In other words, it’s 90°. Right angles follow special rules, as do a few others. [ACT Spoiler Warning] These angles are tested frequently, so memorizing these values is probably a really good idea.
Here’s a “cheat sheet” to help you out:
Some Helpful Hints
Let’s revisit our lovely little triangle:
You should know that you can do this:
This is what’s called the law of sines. Usually, if you have to use this formula, the question will give it to you, but it’s a handy tool to have in your pocket.
Next up is a nifty little equation that you can use on any angle. We’ll follow mathematical convention here and use the symbol θ (pronounced “theta”) to stand in for the value of the angle.
To translate from math back into English, the sine of any angle, squared, plus the cosine of any angle, squared, equals 1. Could be useful if you’re trying to figure out a tough problem on test day, no? If you see this equation anywhere on your math test, just remember that it’s equal to 1.
And to round out our helpful hints, here’s one last equation for you:
Translation: The tangent of any angle equals the sine of the angle divided by the cosine. So if a problem ever asks you to divide the sine by the cosine, you can just plug the tangent right in! (And you can figure out the value of the tangent by using SOHCAHTOA!) Easy!
With all of this in mind, let’s do a sample problem!
The correct answer is… D! Let’s walk through it.
To find the sine of ∠A, you need to know the values of the opposite side (line BC) and the hypotenuse (line AC). You know the hypotenuse is 8, but the problem didn’t give you a value for line BC. It did give you line AB, though, which is 6. So we can use the Pythagorean Theorem to figure out line BC!
is the Pythagorean Theorem, as you might recall from the review on triangles. Substitute in the values we know, and it becomes:
Now that we know the value of line BC, we can figure out the sine of ∠A.
And we have our answer!