Trig is definitely the most intimidating math to most ACT students – that’s because most of them haven’t seen it before (or if they have, usually only in a cursory way). The good news: if you can memorize 1 acronym, 2 formulas, and 1 definition, you’ll be all set to tackle even the hardest problems! First, the acronym: SOHCAHTOA. The three basic trig identities are Sine, Cosine, and Tangent. SohCahToa helps us remember what they are in a right triangle:
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
In plain English, they mean that if you are looking for the “sine” of a certain angle for example, you would divide the length of side opposite that angle by the length of the hypotenuse of the triangle. It’s important to remember that the “opposite” and “adjacent” sides change depending on which angle you are using, so always think of it from the point of view of the angle.
You will definitely encounter questions that require you to use SOHCAHTOA, and you may encounter questions that ask about reciprocal trig identities. Each of the three basic trig identities has a corresponding reciprocal trig identity:
Cosecant = Hypotenuse / Opposite
Secant = Hypotenuse / Adjacent
Cotangent = Adjacent / Opposite
Notice how Sine and Cosecant are the same except the numerator and denominator is flipped. That’s what we mean by reciprocal. It’s easy to remember that “tangent” and “cotangent” are reciprocals since they sound so much alike, but how what about the other two? I once had a math teacher who used, “Co-co no go” as a mnemonic device to help my high school class remember. What he meant was that your brain may think that “cosine” and “cosecant” are reciprocals since they both begin with the prefix “co-“ but that isn’t true. “Sine” goes with “cosecant” and “cosine” goes with “secant.”
Now that we have the acronym down, let’s look at the equations:
The second equation is referred to as the law of sines (where a, b, and c are the sides of the triangle and A, B and C are the opposite angles). Usually if either of these is required on the ACT, the question will give you the formula, but it might be a good idea to memorize them anyway.
Finally, an unusual definition to learn! A radian is another way of measuring an angle. Some harder problems on the ACT will use radians instead of degrees.
There are 2π radians in one circle. Each point on a circle corresponds to a certain number of radians.
This is used in Trig to determine the location of the right triangle (and thus the negative or positive values of the sides). For example, if a Trig question told us that angle theta is between 3π/2 and 2π, we know that the angle must be in the 4th quadrant of the circle.
Look out for questions in which you can use these trig formulas on Test Day! They may seem a little scary now, but after a dozen or so Trig problems, you’ll be amazed at how fast you can solve them!