Many trigonometry mistakes are not random. They repeat in recognisable families: the wrong side is labelled, the wrong ratio is chosen, the calculator is in the wrong mode, the inverse function is missed, or a plausible-looking answer survives because no reasonableness check is made.
This error atlas treats mistakes as diagnostic evidence. The goal is not merely to correct one answer. It is to identify which part of the trigonometric process failed, then repair that mechanism before adding more questions.
Error family 1: wrong side labels
The hypotenuse is opposite the right angle. Opposite and adjacent are labelled relative to the acute angle being used. If the reference angle changes, opposite and adjacent can switch roles.
Diagnostic sign: the student uses a correct trig formula but inserts the wrong side lengths.
Repair: circle the reference angle, mark the right angle, identify the hypotenuse first, and only then label opposite and adjacent.
Error family 2: wrong ratio
Sine connects opposite and hypotenuse. Cosine connects adjacent and hypotenuse. Tangent connects opposite and adjacent.
Diagnostic sign: the side labels are correct, but the chosen function does not contain the two sides that are known and required.
Repair: hide SOHCAHTOA temporarily and ask only: “Which two side roles are involved?” Then choose the ratio that contains exactly those two.
Error family 3: wrong calculator mode
A degree question entered in radian mode can produce a neat decimal that is mathematically unrelated to the intended problem.
Diagnostic sign: the written setup is correct but the numerical value is wildly inconsistent with the diagram.
Repair: make angle mode part of the pre-calculation checklist. For ordinary school geometry questions, confirm degree mode unless another unit is explicitly stated.
Error family 4: direct function instead of inverse function
If sin θ = 0.6 and the question asks for θ, the learner needs sin⁻¹(0.6), not sin(0.6).
Diagnostic sign: the student has formed the correct ratio but produces a tiny decimal instead of an angle.
Repair: distinguish the direction of the task. Trig functions turn angles into ratios; inverse trig functions turn ratios back into angles.
Error family 5: incorrect rearrangement
From tan 35° = h/12, the correct rearrangement is h = 12 tan 35°. From tan 35° = 12/h, the correct rearrangement is h = 12/tan 35°.
Diagnostic sign: the ratio choice is correct but the unknown ends up in the wrong place after rearranging.
Repair: write the equation as a fraction relationship and use ordinary algebra. Avoid memorising separate “triangle tricks”.
Error family 6: wrong triangle
Composite diagrams may contain several right triangles. The required length or angle might not belong to the first visible triangle.
Diagnostic sign: the student uses valid trigonometry on a triangle that does not contain the target quantity.
Repair: highlight the required quantity and identify the smallest right triangle containing both the target and enough known information.
Error family 7: missing prerequisite
Some “trigonometry errors” are really algebra, Pythagoras, parallel-line geometry or unit-conversion errors.
Diagnostic sign: the student can choose the trig ratio correctly but becomes unstable when rearranging equations, finding a missing side first, or interpreting a diagram.
Repair: step backward to the earliest broken prerequisite instead of drilling more trig questions.
Error family 8: angle of elevation or depression measured from the wrong line
Elevation and depression are measured from horizontal reference lines. Measuring from a vertical wall changes the angle and therefore the ratio.
Repair: draw the observer’s horizontal line before marking the angle of sight.
Error family 9: premature rounding
Rounding an intermediate length can make a later angle or distance drift away from the accurate value.
Repair: keep exact values or full calculator precision through intermediate steps, then round once at the end according to the question.
Error family 10: no reasonableness check
An answer can be numerically tidy and geometrically impossible. A calculated leg cannot exceed the hypotenuse in a right triangle. A very small acute angle should not usually produce an opposite side vastly larger than the adjacent side.
Repair: ask three questions before accepting the answer: Is the unit correct? Is the size plausible? Does the result fit the shape of the diagram?
Worked diagnostic 1
A right triangle has angle 30°, hypotenuse 10 cm and opposite side x. A student writes cos 30° = x/10.
The error is not arithmetic. The student has chosen cosine for an opposite–hypotenuse relationship. The repair is side-role selection: sine uses opposite and hypotenuse.
sin 30° = x/10
x = 5 cmWorked diagnostic 2
A student correctly writes tan θ = 3/4 but then calculates tan(0.75) and reports 0.93°.
The ratio is correct. The failure is direction: the task is to recover an angle from a ratio.
θ = tan⁻¹(3/4)
θ ≈ 36.9°Worked diagnostic 3
A student writes sin 40° = x/8, then obtains x = 17.9 cm.
That result is impossible if 8 cm is the hypotenuse, because a leg cannot be longer than the hypotenuse. The setup suggests x should be 8 sin 40°, which is about 5.14 cm. A reasonableness check would have exposed the rearrangement error immediately.
The repair protocol
- Identify the right triangle.
- Circle the reference angle.
- Label H, O and A.
- Select the ratio using the two relevant side roles.
- Write the equation before touching the calculator.
- Rearrange using ordinary algebra.
- Check degree/radian mode.
- Calculate with sufficient precision.
- Check size, unit and geometry.
- Test one changed example to see whether the repair transfers.
Mini transfer test
- Angle 25°, adjacent side 14 cm. Which trig ratio should find the opposite side?
- Opposite side 6 cm, hypotenuse 10 cm. Which inverse function should find the angle?
- A calculated leg is 18 cm while the hypotenuse is 12 cm. What does that tell you?
- An angle of depression is marked from a vertical line. What is wrong?
Answers
1. Tangent. 2. sin⁻¹. 3. The result or setup is wrong because a leg cannot exceed the hypotenuse. 4. Depression must be measured from a horizontal reference line.
Connected routes
Use Choosing Sine, Cosine or Tangent Without Guessing for ratio selection, Angles of Elevation and Depression for diagram translation, and Pythagoras’ Theorem for right-triangle length structure.
Return to the Mathematics Learning Hub for the wider Secondary Mathematics route. The purpose of an error atlas is not to label a student. It is to make the earliest repairable failure visible.