Basic Trig Identities — Strategy & Common Mistakes

Below is a focused strategy guide for Basic Trig Identities on the ACT math section. Use sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ to simplify expressions. The advice here is specific to this topic and to the broader Trigonometry category — the patterns that matter, the mistakes that cost the most points, and the sequence to follow when this question type shows up on test day.

Mindset for this topic

ACT Trigonometry is a small but critical portion (roughly 7%). Top scorers own SOH-CAH-TOA, the Pythagorean identity, and the laws of sines and cosines. If those are automatic, you are done.

For Basic Trig Identities specifically, that mindset translates into one rule: read the prompt twice before you write a single symbol. Most students who miss this question type miss it because they jumped to a calculation before they understood what was being asked. The ACT loves to phrase a Basic Trig Identities question in a way that sounds like a different topic — a word problem, a diagram, a function in disguise. Slow on the read, fast on the math.

The four most common mistakes on Basic Trig Identities

1. Mixing up sine and cosine (opposite vs. adjacent).
2. Computing in degrees when the question is in radians, or vice versa.
3. Forgetting that sin² θ + cos² θ = 1 — this identity solves a surprising number of "find the other ratio" questions in one line.
4. Mis-applying the law of cosines because of a sign error.

If you keep a personal "leak list" (we recommend it on the study tips page), add the one mistake from the list above that you have made most recently. Re-read it before every practice session for two weeks; you will stop making it.

The recommended workflow

Identify what you know (which sides, which angles). Pick the right ratio or law. Compute. Convert units if needed.

Run this workflow on every Basic Trig Identities question you attempt. The first ten times it will feel slow. By the twentieth question it will be automatic, and your accuracy on this topic will jump by 10–20 points worth of raw score.

Pacing

50–80 seconds per trig question. On a ACT that mixes question types, you do not want to be the student who spent 3 minutes on a Basic Trig Identities question and ran out of time on questions you would have nailed.

What to do next

Open the main Basic Trig Identities topic page and re-attempt the six worked questions with this strategy in mind. If you still miss any, use the Basic Trig Identities cheatsheet as a one-page reminder, and revisit our Trigonometry formula sheet until the formulas are automatic.