ACT Math · Trigonometry
Trigonometry on the ACT: Sin, Cos, and Tan — Every Rule Explained
Concepts covered in this guide
| Concept | Named Method | Frequency |
|---|---|---|
| SOH-CAH-TOA: the three ratios, defined and assigned correctly | The Angle-First Label | Very High |
| Setting up and solving a trig equation from a triangle | The Trig Equation Setup | Very High |
| Trig ratios as fractions vs. actual side lengths | The Scale-Up Method | High |
| Quadrant-based sign rules: when sin, cos, and tan are negative | The ASTC Quadrant Map | Medium |
| Calculator mode errors: degree vs. radian and how to check | The Mode Check | Medium |
| Reading trig answer choices in ratio form | The Ratio-Match Method | High |
Concept 1
SOH-CAH-TOA: The Three Ratios, Defined and Assigned Correctly
Very High FrequencySine, cosine, and tangent are ratios between pairs of sides in a right triangle, defined relative to a specific reference angle. The labels “opposite,” “adjacent,” and “hypotenuse” are assigned after identifying which angle the question asks about — they shift when the reference angle shifts. This shifting is the source of the most common ACT trig error.
⚠ ACT trap — labeling sides from the wrong angle
Named Method
The Angle-First Label
Step 1: Circle the reference angle in the problem. This is the angle whose sine, cosine, or tangent the question asks about. Step 2: Label the three sides relative to that angle only. Opposite = the side directly across the triangle from the circled angle. Adjacent = the side touching the circled angle that is not the hypotenuse. Hypotenuse = always the longest side, opposite the right angle.
SOH: sin(θ) = Opposite / Hypotenuse
CAH: cos(θ) = Adjacent / Hypotenuse
TOA: tan(θ) = Opposite / Adjacent
Quick check: opposite and adjacent swap when you switch from one acute angle to the other. If sin(A) = 5/13, then cos(B) = 5/13 (the complementary angle relationship). This is a fast verification tool on ACT questions that give both angles.
ACT-style practice question
Right triangle ABC has a right angle at C. Side BC = 5, side AC = 12, and hypotenuse AB = 13. What is sin(A)?
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Concept 2
Setting Up and Solving a Trig Equation from a Triangle
Very High FrequencyWhen the ACT gives a right triangle with a known angle and a known side, it is asking you to find an unknown side using a trig ratio. The setup is always the same: choose the ratio that connects the known side to the unknown side, write the equation, and solve algebraically. The choice of which ratio to use is determined entirely by which sides are involved — known and unknown — relative to the reference angle.
Named Method
The Trig Equation Setup
Step 1: Apply the Angle-First Label to identify which side is opposite, adjacent, and hypotenuse relative to the given angle. Step 2: Identify which side is known and which is unknown. Step 3: Select the ratio that contains both sides: if the known and unknown are opposite and hypotenuse, use sine. If adjacent and hypotenuse, use cosine. If opposite and adjacent, use tangent. Step 4: Write the equation: trig function(angle) = known/unknown or unknown/known. Step 5: Solve for the unknown by cross-multiplying or isolating the variable.
The equation always takes the form: ratio = (one side) / (another side). If the unknown is in the denominator, multiply both sides by the unknown and then divide by the trig value. If the unknown is in the numerator, multiply both sides by the denominator.
If sin(A) = 3/5: 3/5 = BC/15
Cross-multiply: BC = 15 × (3/5) = 9
ACT-style practice question
Right triangle ABC has a right angle at C. cos(A) = 3/5. The hypotenuse AB = 10. What is the length of the side BC?
Concept 3
Trig Ratios as Fractions vs. Actual Side Lengths
High FrequencyWhen sin(θ) = 4/5, the numbers 4 and 5 are not necessarily the actual lengths of any side in the triangle. They are the ratio between the opposite side and the hypotenuse — meaning the opposite is 4 units for every 5 units of hypotenuse. The actual side lengths could be 4 and 5 in a specific triangle, or 8 and 10, or 40 and 50. The ratio is a proportion, not a measurement, and confusing these two is a reliable source of wrong answers on the ACT.
Named Method
The Scale-Up Method
When a trig ratio is given and an actual side length is also given, find the scale factor by comparing the given side to the ratio’s corresponding component, then multiply all ratio values by the scale factor.
Example: sin(θ) = 3/5 and the hypotenuse is 20. The ratio says hypotenuse corresponds to 5. Scale factor = 20 ÷ 5 = 4. Opposite side = 3 × 4 = 12. Adjacent side (from Pythagorean or by completing the ratio) = 4 × 4 = 16.
When no actual side is given — only the ratio — use the ratio’s numerator and denominator as the side lengths directly (the simplest triangle with that ratio). The ACT explicitly designs some questions where the ratio values and side values coincide and others where they do not. Reading the problem for an actual measurement is the triage step.
✓ Ratio matches actual sides
✗ Ratio ≠ actual sides (scaled)
ACT-style practice question
In a right triangle, sin(θ) = 3/5. The hypotenuse of the triangle is 20. What is the length of the side opposite angle θ?
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Concept 4
Quadrant-Based Sign Rules: When Sin, Cos, and Tan Are Negative
Medium FrequencyIn a right triangle, all trig ratios are positive because side lengths are positive. But the ACT also tests trig in broader contexts where an angle can be in any of the four quadrants of the coordinate plane, and the signs of sin, cos, and tan depend on which quadrant the angle occupies. Harder ACT questions give a quadrant constraint (e.g., “180° < θ < 270°”) along with one trig value and ask for another.
Named Method
The ASTC Quadrant Map
Memory device: All Students Take Calculus — reading quadrants counterclockwise from Q1: All, Sine, Tangent, Cosine. The letter tells you which function(s) are positive in that quadrant; all others are negative.
Step-by-step for harder questions: (1) Identify the quadrant from the given angle range. (2) Note which functions are positive there. (3) Use the given trig ratio to find the magnitudes of the sides using the ratio definition. (4) Assign the correct signs based on the quadrant. sin is linked to y (positive above x-axis), cos is linked to x (positive to the right of y-axis), tan = sin/cos (positive when both have the same sign).
ACT application: if the question gives sin(θ) = −4/5 and says θ is in Q3, then both sin and cos are negative in Q3. Use the magnitude 4/5 to find the sides (opp magnitude = 4, hyp = 5, adj magnitude = 3 via 3-4-5), then apply the negative sign to both sin and cos: sin(θ) = −4/5 (given), cos(θ) = −3/5 (negative in Q3).
ACT-style practice question
Angle θ satisfies 180° < θ < 270° and sin(θ) = −4/5. What is the value of cos(θ)?
Concept 5
Calculator Mode Errors: Degree vs. Radian and How to Check
Medium FrequencyThe ACT expresses all angles in degrees. A calculator set to radian mode will produce completely wrong numerical answers when computing trig functions of degree inputs — and the wrong answers will look plausible enough to select. Every student who uses a calculator on ACT trig questions must verify the calculator is in degree mode before beginning, and again after any time the calculator has been reset or shared.
Named Method
The Mode Check
How to check on a TI-84: Press MODE. On the third row, look for RADIAN and DEGREE. The highlighted option is the active mode. Use the arrow keys to select DEGREE and press ENTER. Press 2ND > QUIT to exit. Your calculator is now in degree mode.
How to verify without navigating menus: compute sin(90). If the result is 1, the calculator is in degree mode. If the result is anything other than 1 (sin(90 radians) ≈ 0.894), the calculator is in radian mode. This single test takes 5 seconds and confirms the mode before any trig work begins.
When this matters on the ACT: any question where you evaluate sin, cos, or tan of a specific numerical angle (e.g., cos(60°), tan(45°), sin(30°)) requires degree mode. Questions that give a trig ratio and ask you to solve algebraically (without pressing the sin/cos/tan buttons) are not affected by calculator mode.
Common mode-error pattern: sin(30) in radian mode ≈ −0.988, not 0.5. cos(60) in radian mode ≈ −0.952, not 0.5. If your calculator returns a negative value for a trig function of an acute angle, mode error is the first thing to check.
✓ Degree mode — correct results
✗ Radian mode — wrong results
ACT-style practice question
A student uses a calculator to find cos(60°) and gets approximately −0.952. Which of the following best explains the student’s error?
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Concept 6
Reading Trig Answer Choices in Ratio Form
High FrequencyMany ACT trig questions present answer choices in ratio form — expressions like BC/AB, AC/BC, or AB/AC — rather than as numerical values. These questions are asking you to identify which ratio of sides equals the requested trig function. No calculator is needed. The skill is correctly assigning opposite, adjacent, and hypotenuse relative to the reference angle and then matching that assignment to the side labels in the answer choices.
Named Method
The Ratio-Match Method
Step 1: Apply the Angle-First Label to identify which labeled side (AB, BC, AC) corresponds to opposite, adjacent, and hypotenuse for the reference angle. Step 2: Write out the requested trig definition in word form: “sine = opposite/hypotenuse.” Step 3: Replace “opposite” and “hypotenuse” with the actual side labels from step 1. Step 4: Match to the answer choices.
This process takes under 30 seconds and is entirely mechanical. The only variable is correctly identifying which side is opposite and which is adjacent for the specific reference angle — which is exactly the Angle-First Label step. Students who skip the angle-first step will match the ratio to the wrong angle and select a plausible but incorrect answer choice.
ACT-style practice question
Right triangle ABC has a right angle at C. Side AB is the hypotenuse. Which of the following expressions equals tan(B)?
Quick-Reference Summary: All 6 ACT Trig Concepts
| Concept | Named Method | Key Rule |
|---|---|---|
| SOH-CAH-TOA and side labeling | The Angle-First Label | Label opposite/adjacent/hypotenuse relative to the reference angle first. Sides shift when the angle shifts. |
| Setting up a trig equation | The Trig Equation Setup | Choose ratio connecting known and unknown sides. Write equation. Cross-multiply to solve. |
| Ratio vs. actual side lengths | The Scale-Up Method | sin(θ) = 3/5 is a proportion. Find scale factor from actual hyp ÷ ratio hyp. Multiply all sides. |
| Quadrant sign rules | The ASTC Quadrant Map | ASTC: All, Sin, Tan, Cos positive in Q1, Q2, Q3, Q4. All others negative. Apply sign after finding magnitude. |
| Calculator mode | The Mode Check | Verify degree mode: compute sin(90). Result must be 1. If not, switch to DEGREE in MODE settings. |
| Ratio-form answer choices | The Ratio-Match Method | Label sides from reference angle first. Write ratio in words. Replace with actual side labels. Match to choices. |
How to Approach Trig Questions on Test Day
Tip 1
Before writing any ratio, circle the reference angle in the problem. Then physically point to each side of the triangle and say which label it gets: opposite, adjacent, or hypotenuse relative to that circled angle. This 10-second habit prevents the most common ACT trig error — assigning sides from the wrong angle. It sounds slow, but incorrect side labeling costs far more time through wrong answers and re-working than correct labeling costs upfront.
Tip 2
Check your calculator mode before the first trig question, not after you get a suspicious answer. In practice, this means pressing sin(90) as a quick test the moment you open a trig problem. The result should be exactly 1. If it is not 1, switch to degree mode before doing anything else. This 5-second check, done once per test section, eliminates an entire category of wrong answers.
Tip 3
When the answer choices are ratios of side labels (AC/BC, BC/AB, etc.) rather than numbers, do not use your calculator at all. These questions are testing pure ratio identification, not computation. Apply the Ratio-Match Method: label sides from the reference angle, write the definition in word form, substitute the actual side labels, and match. Calculator use on these questions wastes time and does not help.
Tip 4
On quadrant sign questions, determine the sign of the answer before computing its magnitude. Check the ASTC map, identify the quadrant, note which functions are positive, and decide whether your final answer will be positive or negative. Then find the magnitude of the ratio from the given trig value. Applying the sign last — after the magnitude is found — prevents sign errors that arise from trying to track positives and negatives simultaneously through the calculation.
Common Questions About ACT Trigonometry
In Quadrant III, only tangent is positive — both sine and cosine are negative. The ASTC Quadrant Map (All, Sine, Tangent, Cosine) tells you which functions are positive in each quadrant: Q1 all positive, Q2 sine only, Q3 tangent only, Q4 cosine only. Everything else in that quadrant is negative.
The reason: sine corresponds to the y-coordinate of a point on the unit circle and cosine corresponds to the x-coordinate. In Q3, both x and y are negative, so both sine and cosine are negative. Tangent = sine/cosine = negative/negative = positive. So even though you are told tan(θ) — and it is positive — you know sin(θ) is negative because y is negative in Q3. Apply this: find the magnitude of sin(θ) from the given tan ratio (using the side relationships), then attach a negative sign.
Q3: sin negative, cos negative, tan positive (negative ÷ negative) Given: tan(θ) = 3/4 in Q3 → sin(θ) = −3/5, cos(θ) = −4/5Even without a diagram, the problem always tells you which two sides are involved. It will name or describe the sides: the height of a building (opposite), the horizontal distance (adjacent), the line of sight (hypotenuse), or something similar. Identify which two sides you have or need, then choose the function that connects those two sides.
The matching is: if you have opposite and hypotenuse (or need one of them and have the other), use sine. If you have adjacent and hypotenuse, use cosine. If you have opposite and adjacent, use tangent. This decision is independent of whether a diagram exists — it depends only on which sides are in play. If you are truly unsure which sides are involved, draw the triangle yourself from the problem description and label it.
Opposite & Hypotenuse → SIN Adjacent & Hypotenuse → COS Opposite & Adjacent → TANThe most common cause is radian mode. Apply the Mode Check immediately: compute sin(90) on your calculator. If the result is 1, you are in degree mode and the calculator is not the problem. If the result is anything else, switch to degree mode (MODE → highlight DEGREE → ENTER) and recompute your answer.
If the calculator is in degree mode and the answer still seems wrong, check two things: first, make sure you entered the angle value, not a side length, as the input to the trig function. Second, make sure you used the correct trig button (sin vs. cos vs. tan) for the ratio the problem sets up. A wrong trig function produces a plausible-looking decimal that is close enough to almost seem right, but it will match a wrong answer choice rather than the correct one.
You do not need to solve to a number when the answer choices are ratios of side labels. These questions test identification, not computation. Apply the Ratio-Match Method: label the triangle’s sides relative to the reference angle, write the definition of the requested trig function in word form, replace the words with the actual side labels, and match to the answer choices.
Example: tan(B) in a right triangle ABC with right angle at C. Label sides from angle B: opposite = AC, adjacent = BC, hypotenuse = AB. tan(B) = opposite/adjacent = AC/BC. Scan the choices for AC/BC. The process takes about 20 seconds and requires no computation at all. The most common error is using angle A’s labeling for angle B’s question — which is exactly why the Angle-First Label step is the mandatory first move.
It matters significantly, and the ACT tests it directly. sin(θ) = 4/5 is a proportion — it says the opposite side is 4 units for every 5 units of hypotenuse. The actual side length depends on the actual size of the triangle. If the hypotenuse is 5, then the opposite side is 4. If the hypotenuse is 25, the opposite side is 20. If the hypotenuse is 1, the opposite side is 0.8.
The ACT tells you the actual hypotenuse (or another actual side) in the problem. Apply the Scale-Up Method: divide the actual side by the corresponding ratio component to find the scale factor, then multiply all ratio components by that scale factor to find the actual side lengths. The ratio alone tells you the shape; the actual measurement tells you the size. Both pieces of information are required to find an actual length.
sin(θ) = 4/5, actual hyp = 15 Scale factor = 15 ÷ 5 = 3 Actual opposite = 4 × 3 = 12 (not 4)