Opening subject page...
Loading your content
Master sine, cosine, and tangent to unlock the relationships between angles and sides in any triangle.
Long before anyone coined the word "trigonometry," ancient civilizations needed a reliable way to connect angles to measurable distances. Egyptian builders calculated the slopes of pyramid faces, Babylonian astronomers tracked planetary motion across the night sky, and Greek navigators charted courses across the Mediterranean. Each group discovered the same core idea independently: if you know a triangle's angles, you can predict its side lengths, and vice versa. That insight is the heartbeat of trigonometry, and it remains one of the most tested topics on the ACT Math section.
The central question that drove all of this work is deceptively simple: Given a triangle with certain known measurements, how can we find the unknown sides and angles? The ACT tests your ability to answer exactly this question using the trigonometric ratios and identities we will explore in the sections ahead.
Trigonometry is built on the idea that a right triangle (a triangle with one 90° angle) has fixed ratios between its sides for any given acute angle. Once you memorize the three primary ratios—sine, cosine, and tangent—you can solve almost every trig problem the ACT throws at you. These ratios are collectively remembered with the mnemonic SOH-CAH-TOA, and they form the foundation for everything else in this lesson.
Notice that the labels opposite and adjacent depend entirely on which acute angle you are considering. If you shift your attention to the other acute angle in the triangle, the opposite and adjacent sides swap. The hypotenuse, however, never changes because it is always the side across from the right angle. This distinction is the number-one source of careless errors on the ACT, so practice identifying the correct sides before you plug anything into a formula.
The ACT expects you to know the three primary trig ratios, the Pythagorean identity, and how to convert between degrees and radians. Below are the key equations you should commit to memory, along with explanations of each variable.
While SOH-CAH-TOA works perfectly for right triangles, the unit circle extends trigonometry to angles of any size—including those greater than 90° and negative angles. The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. Every point on the circle can be written as (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. The ACT sometimes asks questions about trig values in different quadrants, so understanding how signs change around the circle is essential.
| Angle (°) | Angle (rad) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
Let's walk through a typical ACT-style problem step by step. Suppose you're given a right triangle where one acute angle is 35° and the hypotenuse measures 20 cm. The question asks you to find the length of the side opposite the 35° angle.
SOH-CAH-TOA is incredibly effective for right triangles, but trigonometry on the ACT goes a bit further. Understanding where each tool shines—and where it falls short—will help you choose the fastest approach on test day.
| Tool / Concept | Strengths | Limitations |
|---|---|---|
| SOH-CAH-TOA | Fast and direct for any right triangle. Only requires one angle and one side to find another side. | Only applies to right triangles. Cannot be used directly for obtuse or non-right triangles. |
| Unit Circle | Handles any angle (including > 90°, negatives, and radians). Essential for determining signs in different quadrants. | Requires memorization of key coordinates. Does not directly give side lengths of a specific triangle. |
| Pythagorean Identity | Lets you find sin from cos (or vice versa) without a calculator. Useful when the ACT gives you one ratio. | Only connects sin and cos. You still need the tangent identity or other information to find tan. |
| Law of Sines / Law of Cosines | Extends trig to any triangle, not just right triangles. Occasionally tested on the ACT. | More complex setup. Requires careful attention to which sides and angles correspond. |
The ACT occasionally dips into topics that go beyond basic right-triangle trig. The table below shows how the foundational concepts you've learned connect to these more advanced ideas. You don't need to master all of them, but recognizing these connections can help you handle the hardest questions on the test.
| Foundational Concept | Advanced Extension | When It Appears on the ACT |
|---|---|---|
| SOH-CAH-TOA (basic ratios) | Reciprocal functions: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ | Rarely tested; may appear in the hardest 5 questions. |
| Unit circle (sin and cos of standard angles) | Graphing y = sin x and y = cos x; amplitude, period, and phase shifts | 1–2 questions per test on recognizing or interpreting trig graphs. |
| sin²θ + cos²θ = 1 | Double-angle identities: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ | Occasionally tested; may appear as simplification or substitution problems. |
| Right-triangle applications | Law of Sines: a/sin A = b/sin B; Law of Cosines: c² = a² + b² − 2ab cos C | Tested 0–1 times per exam, but can be a scoring differentiator. |
If you're aiming for a score in the 30+ range, spend some time familiarizing yourself with trig graphs and the Law of Sines and Cosines. These topics bridge the gap between geometry-level trig and the precalculus material you'll encounter in college. Even if these advanced questions appear only once or twice, they are often worth the same number of points as easier questions, and mastering them can set you apart.
Trigonometry connects angles to side lengths through three fundamental ratios: sine (opposite / hypotenuse), cosine (adjacent / hypotenuse), and tangent (opposite / adjacent). The mnemonic SOH-CAH-TOA is your go-to tool for right-triangle problems, while the unit circle extends these ratios to angles of any size by defining every point on a circle of radius 1 as (cos θ, sin θ).
Key formulas to remember include the Pythagorean identity (sin²θ + cos²θ = 1) and the relationship tan θ = sin θ / cos θ. Memorize the trig values of special angles (30°, 45°, 60°) and know the sign rules for each quadrant using "All Students Take Calculus." On the ACT, always identify which sides and angles you have before choosing a ratio, and double-check that your answer is reasonable by confirming that no leg exceeds the hypotenuse.