Trigonometry on the ACT: Complete Study Guide

The ACT tests more trigonometry than the SAT — typically 4 to 6 questions out of 60 on the math section, ranging from straightforward SOHCAHTOA to harder law of sines/cosines and trig graph problems. These questions appear mainly in the later, harder portion of the test (questions 40 through 60), so they tend to be worth a bit more effort to prepare for. The good news: ACT trig is predictable. The same handful of concepts show up year after year, and with targeted practice you can pick up every one of those points.

What the ACT Tests in Trigonometry

The ACT covers a broader range of trig topics than the SAT. Here is the full list, roughly ordered from most to least common:

• SOHCAHTOA — Finding sine, cosine, and tangent of an angle in a right triangle (easy to medium difficulty). See SOH CAH TOA for a full walkthrough.
• Special right triangles — Side ratios for 30-60-90 and 45-45-90 triangles, often tested alongside basic trig ratios.
• Unit circle values — Exact values for $\sin$, $\cos$, and $\tan$ at standard angles ($0°$, $30°$, $45°$, $60°$, $90°$, and their multiples). See Unit Circle.
• Trig identities — The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$, reciprocal identities ($\csc$, $\sec$, $\cot$), and cofunction identities ($\sin\theta = \cos(90° - \theta)$). See Trig Identities.
• Law of sines and law of cosines — Solving oblique (non-right) triangles. This is an ACT topic that does not appear on the SAT. See Law of Sines and Law of Cosines.
• Graphing trig functions — Identifying amplitude, period, and phase shift from equations or graphs. Also ACT-only. See Graphs of Sine and Cosine.
• Radian-degree conversion — Converting between the two angle systems using $180° = \pi$ radians. See Radians and Degrees.
• Trig equations — Solving simple equations such as $\sin(x) = 0.5$ for specific angle values.

What is generally NOT on the ACT: sum and difference formulas, proving identities step by step, half-angle formulas, and double-angle formulas. You may have studied these in a trig class, but you can safely deprioritize them for the ACT.

Key Differences from the SAT

If you are deciding where to focus your study time, it helps to know how the two tests differ on trig:

The ACT casts a wider net across trig topics but tends to ask each question in a fairly direct way. The SAT uses fewer trig topics but may wrap them in multi-step problems. If you are preparing for the ACT specifically, breadth matters more than depth.

ACT-Style Practice Problems

Work through these seven problems, then click to check your solution. Each one mirrors a real ACT question type.

ACT Trig Strategy Tips

Knowing the math is only half the battle — you also need to manage your time well on a test that gives you just one minute per question on average.

1. Do the easy SOHCAHTOA questions first. Every question on the ACT is worth the same number of points. A basic right-triangle ratio you solve in 30 seconds earns the same score as a law-of-cosines problem that takes you three minutes. Get the easy points banked early.

2. Memorize the graphing formulas. For $y = A\sin(Bx - C) + D$: amplitude $= |A|$, period $= 2\pi / |B|$, phase shift $= C / B$, vertical shift $= D$. These two lines of memorization can get you a free point on most tests.

3. Know when to use each law. Use the law of sines when you have a matched angle-side pair (AAS or ASA). Use the law of cosines when you have SAS (two sides and the included angle) or SSS (all three sides). If you pick the wrong law, you will waste time.

4. Keep your calculator in the right mode. If the problem gives angles in degrees, your calculator must be in degree mode. If the problem uses radians (look for $\pi$ in the angle), switch to radian mode. A wrong mode setting turns an easy question into a wrong answer.

5. Skip and come back. If a trig question at number 55 has you stuck after 60 seconds, mark it and move on. Do not let one hard trig problem eat the time you need for two or three easier questions elsewhere.

Study Priority for the ACT

If your study time is limited, focus on topics in this order — from highest to lowest payoff:

1. SOHCAHTOA — appears on virtually every ACT. Master right-triangle ratios first.
2. Special triangles and unit circle values — know the exact values for $30°$, $45°$, $60°$, and their radian equivalents.
3. Graphing basics — amplitude and period questions are common and quick to answer once you know the formulas.
4. Law of sines and law of cosines — one or two questions per test. Learn the formulas and practice a few problems.
5. Identities — least commonly tested. Know the Pythagorean identity and reciprocal definitions, but do not spend hours memorizing obscure formulas.

This order maximizes your expected point gain per hour of study. A student who has SOHCAHTOA and graphing down cold will pick up more ACT trig points than one who has memorized every identity but stumbles on basic ratios.

Key Takeaways

• The ACT includes 4 to 6 trig questions, more than the SAT, and covering a broader range of topics
• Most ACT trig is predictable — the same topic types repeat from test to test
• Start your study with SOHCAHTOA and special triangles, then expand to graphing, laws, and identities
• On test day, do easy trig questions first, keep your calculator in the correct mode, and do not let hard questions steal time from easy ones
• With focused practice on the topics above, trig can be one of the most efficient places to gain points on the ACT

For deeper dives into each topic, see the pages below:

• SOH CAH TOA — right-triangle trig fundamentals
• Graphs of Sine and Cosine — amplitude, period, phase shift
• Law of Sines and Law of Cosines — oblique triangle solving
• Trig Identities — Pythagorean, reciprocal, and cofunction identities
• Unit Circle Chart — printable reference for all standard angle values

Return to Trigonometry for the full topic list.