ACT Trigonometry Formula Sheet
SOH-CAH-TOA, identities, and laws of sines & cosines for the ACT Trigonometry reporting category.
Below are the 14 formulas tested in this category, with the situation each one applies to. Drill until you can write the whole sheet from memory in under two minutes — that's the standard top scorers reach by test day.
Each formula links back to the topic page where it's worked into example problems. If a formula doesn't yet feel automatic, click into the topic page and grind that topic's six practice questions.
Formulas
| Concept | Formula | When to use it |
|---|---|---|
| Sine | sin θ = opposite / hypotenuse | SOH. |
| Cosine | cos θ = adjacent / hypotenuse | CAH. |
| Tangent | tan θ = opposite / adjacent | TOA. |
| Reciprocal identities | csc = 1/sin ; sec = 1/cos ; cot = 1/tan | Inverse ratios. |
| Pythagorean identity | sin² θ + cos² θ = 1 | For any θ. |
| Quotient identity | tan θ = sin θ / cos θ | Combines the three. |
| Law of sines | a/sin A = b/sin B = c/sin C | Non-right triangles. |
| Law of cosines | c² = a² + b² − 2ab cos C | Generalises Pythagoras. |
| Sine of common angles | sin 30° = ½ ; sin 45° = √2/2 ; sin 60° = √3/2 | Memorise these. |
| Cosine of common angles | cos 30° = √3/2 ; cos 45° = √2/2 ; cos 60° = ½ | Mirror of sine. |
| Period of sin/cos | 2π (or 360°) | One full wave. |
| Amplitude of y=a sin(bx) | |a| | Vertical stretch. |
| Period of y=a sin(bx) | 2π / |b| | Horizontal compression. |
| Degree ↔ radian | 180° = π rad | Multiply by π/180 or 180/π. |
How to memorise this sheet in a week
Day 1. Read every row out loud. Connect each formula to a worked example by clicking into the topic page on which it appears. Goal: understand why each formula is true, not just memorise it.
Day 2. Cover the right-hand columns and quiz yourself on each concept. Write the formula and the use case from memory. Mark the ones you missed.
Day 3. Re-quiz only the ones you missed yesterday. Keep cycling until your error count hits zero.
Day 4. Re-do the full quiz. You should know every row.
Day 5. Apply each formula in a problem context — pick a topic page from the list below and solve all six questions.
Day 6–7. Sit a real past paper. Note any formulas you blanked on under time pressure and add them back to your daily quiz.
Topics that use these formulas
- Sine, Cosine and Tangent6 Qs
- SOH-CAH-TOA Applications6 Qs
- Basic Trig Identities6 Qs
- Law of Sines6 Qs
- Law of Cosines6 Qs
- Graphs of Trig Functions6 Qs
- Inverse Trig Functions6 Qs
- Solving Trig Equations6 Qs
Common mistakes when applying these formulas
Three errors account for the majority of dropped points on formula-driven questions:
- Substituting before simplifying. Always simplify symbolically as far as you can before plugging numbers in. It cuts arithmetic errors in half.
- Mixing units. If a formula expects metres and you have centimetres, convert first, not at the end. The number you compute is meaningless until the units agree.
- Forgetting the ½ in area or volume formulas. Triangle, trapezoid, sector, cone, pyramid — all have a ½ or ⅓ that's easy to drop. Write the formula on the page before plugging in.