ACT Trigonometry Questions

Trigonometry questions on the ACT test how well you handle right-triangle ratios and radian measure, mixed in with the rest of ACT Math rather than grouped in their own block. You see them inside word problems, geometry figures, and standalone equation questions. ACT folds them into the Geometry reporting category, worth 17 to 20 percent of the 41 scored Math questions, so no official trig-only count exists. The underlying skill set stays narrow: three ratios, a short list of common angle values, and one conversion rule. If ratios and unit conversion feel shaky, review the ACT Math pre-algebra guide first.

This guide covers SOHCAHTOA, the unit circle and radian conversion, the trig identities worth knowing, sine and cosine graphs, and the traps these questions set for you. Each section gives you the rule first, then a worked example. Where ACT's published materials stop short of naming a topic, this guide says so instead of guessing. It ends with links to a free trigonometry drill and full practice tests on 10exam.

How the ACT Weights Trigonometry

Trigonometry does not get its own reporting category on the ACT. Instead, trig ratios and equations of conic sections sit inside the Geometry category, alongside angle relationships, triangle and circle properties, and area and volume problems. The table below shows how Geometry compares to the other subcategories inside Preparing for Higher Math.

Preparing for Higher Math makes up 80 percent of the 41 scored questions on the Math test, with the remaining 20 percent falling under Integrating Essential Skills. ACT does not publish an exact count or percentage of trigonometry-only questions inside the Geometry slice, so treat trig as one of several figure types you need ready rather than a fixed number of guaranteed questions. Check the official ACT content description for the current breakdown before test day.

SOHCAHTOA Basics

SOHCAHTOA is the memory device for the three basic trig ratios in a right triangle. Each ratio compares two of the triangle's three sides relative to a chosen angle. The three letter groups pair every function with its two sides, which removes the guesswork under time pressure. Recite it during practice until the pairing arrives without effort.

Use the table below to match each ratio to the sides it compares.

These ratios only apply to right triangles. Identify the right angle first, then label the hypotenuse as the side opposite it. From your angle of interest, the other two sides are opposite and adjacent.

For example, a right triangle has a hypotenuse of 10 and a leg of 6 opposite a given angle. The sine of this angle is 6 over 10, or 0.6. Since the third side is 8 by the Pythagorean theorem, the cosine of the same angle is 8 over 10, or 0.8, and the tangent is 6 over 8, or 0.75.

Solving for a Missing Side or Angle

Set up a SOHCAHTOA ratio using the angle and side you know, then solve for the missing value. If you know an angle and one side, multiply or divide to find the other side. If you know two sides, use the inverse trig function on your calculator to find the angle.

• Known angle and hypotenuse, find the opposite side: multiply the hypotenuse by the sine of the angle.
• Known angle and adjacent side, find the hypotenuse: divide the adjacent side by the cosine of the angle.
• Known opposite and adjacent sides, find the angle: take the inverse tangent of opposite divided by adjacent.

Unit Circle and Radians

The unit circle is a circle with a radius of 1 centered at the origin. It links angle measure directly to sine and cosine values, since the coordinates of any point on the circle equal cosine of the angle and sine of the angle. Angles are measured counterclockwise from the positive x-axis. Learning the circle gives you exact values for the common angles without touching a calculator.

A full circle spans 360 degrees or 2 pi radians. Convert between the two units with the relationship 180 degrees equals pi radians. ACT's own official sample Math test includes a degree-to-radian conversion question, so this conversion is confirmed within scope.

Use this table for the degree-radian pairs and their sine and cosine values you see most often.

Convert degrees to radians by multiplying the degree value by pi over 180. Convert radians to degrees by multiplying the radian value by 180 over pi. For example, 75 degrees converts to 75 times pi over 180, which simplifies to 5 pi over 12 radians.

Trig Identities

ACT's official content description names right-triangle ratios and radian conversion directly, but it does not itemize a fixed list of trig identities you must memorize. Knowing the identities below still helps you simplify equations faster on questions combining sine, cosine, and tangent. They also give you a second route to an answer when a figure hands you one ratio and asks for another. Treat them as speed tools rather than a guaranteed question type.

These are the identities worth having ready.

• The Pythagorean identity: sine squared of an angle plus cosine squared of the same angle equals 1.
• Tangent equals sine divided by cosine, for the same angle.
• Reciprocal identities: cosecant is 1 over sine, secant is 1 over cosine, and cotangent is 1 over tangent.
• Complementary angle identity: the sine of an angle equals the cosine of its complement, meaning 90 degrees minus the angle.

For example, if sine of an angle equals 0.6, the Pythagorean identity gives cosine squared equals 1 minus 0.36, or 0.64, so cosine equals 0.8 for an angle in the first quadrant. Tangent follows straight from the second identity: 0.6 divided by 0.8, or 0.75. The quadrant caveat is the part students skip, because cosine squared equals 0.64 leaves two roots, positive 0.8 and negative 0.8. Read the question for any restriction on the angle before you pick the positive root, since cosine turns negative in the second and third quadrants.

Graphing Sine and Cosine Functions

ACT's official content description and its published sample questions do not confirm graphs of sine and cosine functions, such as amplitude, period, or transformations, as a tested topic. Check the official ACT content description directly for the current scope before you spend heavy prep time here. Ratios and radian conversion have confirmed coverage, so put your hours there first. The four features below take a few minutes to learn and cover you if a sine or cosine graph does appear.

If a question does show you a sine or cosine graph, these basic features let you read it correctly.

• Amplitude is half the distance between the graph's highest and lowest points.
• Period is the horizontal distance the graph takes to complete one full cycle before repeating.
• A standard sine graph starts at 0, rises to its maximum, returns to 0, drops to its minimum, then returns to 0 to complete one period.
• A standard cosine graph starts at its maximum, drops to 0, reaches its minimum, returns to 0, then climbs back to its maximum.

Common Trig Traps

Trig questions hide their difficulty in setup errors more often than in the math itself. The arithmetic runs short once the ratio is right, so most lost points trace back to the first line of work. Slow down on labeling and calculator settings, and the rest follows. Watch for these traps before you commit to an answer.

• Mixing up opposite and adjacent sides relative to the angle you are using. Relabel the sides for each specific angle in a question with more than one angle marked.
• Forgetting to check whether your calculator is set to degree mode or radian mode. An answer computed in the wrong mode looks like a real number but is wrong.
• Using the wrong ratio because you did not identify the right angle first. SOHCAHTOA only holds for right triangles, so confirm the 90-degree angle before setting up a ratio.
• Rounding a ratio too early in a multi-step problem. Carry exact fractions or extra decimal places until the final step.
• Answer choices include the result of common mistakes, like swapping sine and cosine. Getting a listed answer does not confirm your setup was correct.

Practice Trigonometry on 10exam

Our free Trigonometry drill covers SOHCAHTOA, the unit circle, radian conversion, and trig identities, with instant explanations after each question and no signup required. The easy sets keep you on single-step work: label the sides, pick the ratio, convert one angle. The medium sets add a second step, mixing inverse functions and identity substitutions into short word problems. The hard sets stack multi-triangle figures and exact-value work in radians, which is where timing pressure shows up. Work the tiers in order and read every explanation, including the ones on questions you answered correctly.

For full-length, timed Math practice under real test conditions, take one of our free practice tests on the ACT exam page. Review every missed trig question, then return to the drill above to reinforce the specific skill which tripped you up. See our ACT Math section guide for the full test format, calculator rules, and content breakdown across every reporting category.

Sources

• Preparing for the ACT Test 2026-2027 | ACT