Trigonometry on the ACT is overwhelmingly right triangle trigonometry, and it runs on one memory device: SOHCAHTOA, meaning sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. If the word trig makes you tense up, take a breath. This is a small, rule-based topic, and students who avoid it are usually leaving easy points on the table.
What SOHCAHTOA actually says
Pick one of the two non-right angles in a right triangle. Relative to that angle, the three sides get names. The hypotenuse is across from the right angle, the opposite side is directly across from your chosen angle, and the adjacent side is the remaining leg, the one touching your angle that is not the hypotenuse. Then sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent.
Opposite and adjacent depend on which angle you chose
This is where nearly every trig error comes from. The hypotenuse never changes, but opposite and adjacent swap depending on which acute angle you are working with.
Take a 3-4-5 right triangle. Call the angle across from the side of length 3 angle A, and the angle across from the side of length 4 angle B. For angle A, the opposite is 3 and the adjacent is 4, so sine of A is three fifths, cosine of A is four fifths, and tangent of A is three fourths. For angle B everything flips: four fifths, three fifths, and four thirds.
Same triangle, completely different ratios. Before you write a fraction, put your finger on the angle named in the question and label the sides from there. One sanity check: for an acute angle, sine and cosine are always between 0 and 1, because a leg is shorter than the hypotenuse.
Choosing the ratio
The decision is mechanical. Name the side you were given and the side you want, then pick the ratio holding exactly those two.
- Given the hypotenuse, want the opposite: sine.
- Given the hypotenuse, want the adjacent: cosine.
- Given one leg, want the other leg, no hypotenuse: tangent.
Three worked examples
A right triangle has a hypotenuse of 20 and an angle of 35 degrees. Find the side opposite that angle. You have the hypotenuse and want the opposite, so use sine: sine of 35 degrees equals x over 20, so x equals 20 times the sine of 35 degrees, or about 11.5.
A 12 foot ladder leans against a wall at a 70 degree angle with the ground. How high up the wall does it reach? The ladder is the hypotenuse and the height is opposite the 70 degree angle, so use sine again: 12 times the sine of 70 degrees, or about 11.3 feet. That is a little less than 12, the sanity check working.
You stand 50 feet from the base of a tower and look up at the top with an angle of elevation of 40 degrees. How tall is the tower? The 50 feet is adjacent and the height is opposite, with no hypotenuse involved, so use tangent: 50 times the tangent of 40 degrees, or about 42 feet. If the unknown lands in the denominator, say the tangent of 25 degrees equals 10 over x, divide instead, giving about 21.4.
Often the answer is an expression, not a decimal
Many ACT trig questions never want you to compute anything. The choices read like 20 sin 35 and 20 cos 35, so your only job is the setup. When you do need a decimal, make sure your calculator is in degree mode rather than radian mode, and see the guide to picking a calculator for the ACT if you have not settled on one yet.
What else trig might ask
Right triangles are the core, but a question here or there reaches further: reading the amplitude or period off a sine or cosine graph, or applying an identity such as tangent equals sine over cosine. Those sit near the top of the difficulty range, so master SOHCAHTOA first.
Special right triangles also give exact values without a calculator. The tangent of 45 degrees is 1, because a 45-45-90 triangle has equal legs, and the sine of 30 degrees is one half, from the 30-60-90 ratio of 1 to the square root of 3 to 2. Those connections come up again in how to solve ACT triangle problems.
Traps and a quick self-check
The dominant trap is assigning opposite and adjacent relative to the wrong angle, which produces a real number that happens to be wrong. Next is reaching for sine out of habit when the problem never mentions the hypotenuse. Last is inverting the fraction, so confirm the hypotenuse is on the bottom for sine and cosine.
Try these. In a right triangle with legs 5 and 12 and a hypotenuse of 13, what are the cosine and the tangent of the angle opposite the side of length 5? And if a right triangle has a 60 degree angle with an adjacent leg of 8, which ratio finds the opposite leg? The answers are twelve thirteenths, five twelfths, and tangent, giving 8 times the tangent of 60 degrees. A focused trig set on thirty-six is the fastest way to make the labeling step automatic, and you can see where trig sits in how much geometry is on the ACT.
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