ACT trigonometry accounts for about 3 to 4 questions out of 45 on the math section — roughly 7% of your score. The good news: most of those questions test the same handful of formulas and patterns. This guide covers every trig concept the ACT tests, from basic SOH-CAH-TOA ratios to the law of sines, with worked examples and practice questions so you can turn trigonometry on the ACT into reliable points on test day.
Nearly every ACT trigonometry question starts with the same three ratios. SOH-CAH-TOA is a mnemonic that encodes them: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. If you memorize nothing else for ACT math trigonometry, memorize this.
These three ratios relate the sides of a right triangle to one of its acute angles. Given a right triangle where you know (or can find) the lengths of two sides, you can calculate any trig ratio for either acute angle. The ACT typically gives you a diagram with labeled sides and asks you to express one of these ratios as a fraction.
The key is remembering which sides go in the numerator and denominator. Sine uses the side across from the angle (opposite) over the longest side (hypotenuse). Cosine uses the side next to the angle (adjacent) over the hypotenuse. Tangent uses opposite over adjacent — no hypotenuse involved.
A common mistake is labeling sides by their position in the diagram rather than their position relative to the angle in question. The hypotenuse is always the side across from the 90-degree angle — it is the longest side. The opposite side is directly across from the angle you are working with. The adjacent side is next to the angle (but is not the hypotenuse).
When the ACT asks about a different angle in the same triangle, the opposite and adjacent sides swap. The hypotenuse stays the same. Always start by identifying which angle the question is asking about, then label the sides from that angle's perspective.
The ACT frequently uses special right triangles because they have clean, predictable side ratios. Recognizing them lets you skip the Pythagorean theorem and solve faster. A 45-45-90 triangle has sides in the ratio 1 : 1 : √2. A 30-60-90 triangle has sides in the ratio 1 : √3 : 2 (shortest to longest).
| Triangle Type | Angle Measures | Side Ratios | Example |
|---|---|---|---|
| 45-45-90 | 45°, 45°, 90° | 1 : 1 : √2 | Legs = 5, Hypotenuse = 5√2 |
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 | Short leg = 4, Long leg = 4√3, Hyp = 8 |
| 3-4-5 Triple | ≈37°, ≈53°, 90° | 3 : 4 : 5 | Sides = 6, 8, 10 (scaled ×2) |
| 5-12-13 Triple | ≈23°, ≈67°, 90° | 5 : 12 : 13 | Sides = 10, 24, 26 (scaled ×2) |
Worked Example
In right triangle ABC, angle C is 90° and angle A is the reference angle. The side opposite angle A is 5 and the hypotenuse is 13. Find sin A, cos A, and tan A.
Beyond the big three (sine, cosine, tangent), the ACT occasionally tests their reciprocals. Secondary trig functions appear on a maximum of one question per ACT test, so knowing these definitions can earn you a point that many students leave on the table.
Each reciprocal function is simply the flip of a primary ratio. Cosecant (csc) = 1/sin = hypotenuse/opposite. Secant (sec) = 1/cos = hypotenuse/adjacent. Cotangent (cot) = 1/tan = adjacent/opposite. When you see one of these on the ACT, mentally replace it with the basic ratio, flip the fraction, and you have your answer.
The most important ACT trig identity is sin²θ + cos²θ = 1. If you know one of the two values, you can find the other. For example, if sin θ = 3/5, then cos²θ = 1 − (3/5)² = 1 − 9/25 = 16/25, so cos θ = 4/5 (assuming the angle is in Quadrant I where cosine is positive).
A related identity you should know: tan θ = sin θ / cos θ. This is useful for simplifying expressions or converting between ratios when the ACT gives you an identity-based question.
The unit circle extends trigonometry beyond right triangles. On the ACT unit circle, each point has coordinates (cos θ, sin θ), where θ is the angle measured from the positive x-axis. Understanding this connection between coordinates and trig values is what the harder ACT trig questions test.
Radians are an alternative unit for measuring angles: 360° = 2π radians. To convert degrees to radians, multiply by π/180. To go the other way, multiply radians by 180/π. The ACT may give you an angle in one unit and ask for the other, or require you to interpret a radian-measured graph.
Enter an angle in degrees to convert it to radians \u2014 useful for unit circle and graph problems.
You do not need to memorize the entire unit circle. Focus on five key angles and their sine and cosine values: 0° (sin 0, cos 1), 30° (sin 1/2, cos √3/2), 45° (sin √2/2, cos √2/2), 60° (sin √3/2, cos 1/2), and 90° (sin 1, cos 0). These are the building blocks for every other angle on the unit circle.
The mnemonic "All Students Take Calculus" tells you which trig functions are positive in each quadrant. In Quadrant I, All functions are positive. In Quadrant II, only Sine is positive. In Quadrant III, only Tangent is positive. In Quadrant IV, only Cosine is positive. Once you know the reference angle's values, the ASTC rule tells you the sign.
Worked Example
Convert 150° to radians and determine sin(150°) and cos(150°) using the unit circle.
The ACT occasionally asks you to identify a trig function from its graph or determine properties like amplitude and period from an equation. These questions look intimidating but follow a simple formula once you know what to look for.
For a function in the form y = A sin(Bx + C) + D: the amplitude is |A| (how far the wave stretches above and below its center line), and the period is 2π/|B| (how long it takes for one complete cycle). Cosine follows the same rules. Tangent is different — its period is π/|B| and it has no amplitude (it extends to infinity).
In the equation y = A sin(Bx + C) + D, the value C creates a horizontal (phase) shift: the graph shifts left by C/B units. The value D shifts the entire graph up or down vertically. On the ACT, you are most likely to be asked about amplitude and period. Phase shifts appear less frequently but are worth recognizing if you are aiming for a high score.
Worked Example
Identify the amplitude, period, and vertical shift of y = 3 sin(2x) + 1.
When a triangle is not a right triangle, SOH-CAH-TOA does not apply directly. That is where the law of sines ACT questions and law of cosines come in. These formulas appear on a maximum of one question per ACT test, but knowing them turns a potential skip into a free point.
Use the Law of Sines (a/sin A = b/sin B = c/sin C) when you have at least one angle-side pair and need to find a side opposite a known angle, or an angle opposite a known side. This covers ASA and AAS setups.
Use the Law of Cosines (c² = a² + b² − 2ab·cos C) when you have two sides and the included angle (SAS) or all three sides (SSS) and need to find a missing side or angle. Think of it as the Pythagorean theorem with a correction factor for the angle.
The ACT will always give you enough information to determine which law to use. Start by listing what you know: if you have an angle-side pair, try the Law of Sines first. If you have SAS or SSS, the Law of Cosines is your tool. The question is designed so that the correct law is clear from the given information.
Worked Example
In triangle DEF, angle D = 40°, angle E = 75°, and side d (opposite angle D) = 10. Find side e (opposite angle E).
Knowing the formulas is half the battle. The other half is applying them efficiently under the time pressure of 45 questions in 50 minutes. Here are the most important ACT trig questions strategies and the errors that cost students the most points.
Most ACT trig questions fall into a handful of predictable patterns. The most common type gives you a right triangle diagram and asks for a trig ratio — pure SOH-CAH-TOA. Word problems involving angles of elevation or depression are next most common; these require you to draw a right triangle from the scenario. Less frequent are reciprocal function questions, radian conversions, graph interpretation, and law of sines/cosines problems.
| Question Type | Frequency per Test | Difficulty | Key Skill |
|---|---|---|---|
| SOH-CAH-TOA ratio from a diagram | 1–2 questions | Easy–Medium | Identify opposite/adjacent/hypotenuse and set up ratio |
| Angles of elevation/depression | 0–1 questions | Medium | Draw the right triangle from the word problem |
| Reciprocal trig functions | 0–1 questions | Medium | Know csc, sec, cot as reciprocals |
| Unit circle / radian conversion | 0–1 questions | Medium–Hard | Convert degrees to radians, read coordinates |
| Trig graph identification | 0–1 questions | Hard | Identify amplitude, period, and shifts from y = A sin(Bx + C) + D |
| Law of sines / law of cosines | 0–1 questions | Hard | Choose the correct law and solve for missing side or angle |
Always verify that your calculator is in degree mode before starting the math section. A calculator set to radians will give you wrong answers on every trig calculation. On the TI-84, check the MODE screen; on the TI-Nspire, check the document settings. This two-second check prevents a cascade of errors.
With 45 questions in 50 minutes, you have about 67 seconds per question on average. Trig questions tend to appear in the second half of the test (questions 30-45), where difficulty ramps up. If a trig problem is taking more than 90 seconds, mark it and move on — you can return if time allows.
The ACT will never ask you to find an actual angle measure using inverse trig functions. Questions only require you to express a ratio. Students who waste time trying to compute arcsin or arccos are making the problem harder than it needs to be.
Other common mistakes include drawing diagrams without labeling them (always label sides and angles), forgetting to check which quadrant an angle falls in when using the unit circle, and mixing up the law of sines with the law of cosines. When no diagram is provided, always draw one yourself — even a rough sketch helps you visualize which sides are opposite and adjacent.
| Formula | What It Means | When to Use It |
|---|---|---|
| sin θ = Opposite / Hypotenuse | The ratio of the side opposite the angle to the hypotenuse | Right triangle ratio problems — the most common ACT trig question type |
| cos θ = Adjacent / Hypotenuse | The ratio of the side adjacent to the angle to the hypotenuse | Right triangle ratio problems and simplifying expressions |
| tan θ = Opposite / Adjacent | The ratio of the opposite side to the adjacent side | Right triangle ratio problems, angles of elevation/depression |
| csc θ = 1 / sin θ | Reciprocal of sine | Rare — at most one question per test |
| sec θ = 1 / cos θ | Reciprocal of cosine | Rare — at most one question per test |
| cot θ = 1 / tan θ | Reciprocal of tangent | Rare — at most one question per test |
| sin²θ + cos²θ = 1 | Pythagorean identity linking sine and cosine | Simplifying expressions or finding one ratio from another |
| Radians = Degrees × (π/180) | Converts degrees to radians | Unit circle and graph interpretation questions |
| a/sin A = b/sin B = c/sin C | Law of Sines — relates sides to opposite angles | Non-right triangles when you have an angle-side pair |
| c² = a² + b² − 2ab·cos C | Law of Cosines — generalized Pythagorean theorem | Non-right triangles with SAS or SSS information |
Enter the opposite and adjacent sides to find the hypotenuse and all trig ratios for the reference angle.