Practice maths class test — Year 4: trigonometry.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

The trigonometry test comes in November-December of Year 4, shortly after the September quiz on trigonometric functions, of which it is the natural continuation: there you studied sine and cosine as functions, here you use them to solve any triangle, not just right-angled ones.

The first mistake is about the chord theorem: chord = 2R·sin α, where α is the inscribed angle that sees that chord — not the angle at the centre, which is twice as big. Using the angle at the centre instead of the inscribed angle does not simply double the result, because the angle is inside the sine: the final number is wrong in a way that depends on the angle itself, and in a test this mistake goes unnoticed until the result is absurd (a chord longer than the diameter, for example).

The second is about the area with the trigonometric formula, Area = ½ · a · b · sin γ: γ must be the angle between the two sides a and b that you multiply, not just any angle of the triangle. If the problem gives you the three angles and two sides that are not adjacent, you must first find the third side or the right angle — you cannot drop an angle into the formula at random just because it is the one you know.

The third is practical and costs marks whatever the theory: a calculator set to radians when the exercise is in degrees (or the other way round) gives completely different numbers even though you understood the method perfectly — it is worth checking the mode before you start the test, not after the first result that looks strange.

To prepare, redo one exercise with the chord theorem and one with the trigonometric area, checking explicitly which angle you are using in each formula: that is where mistakes happen, not in the formulas themselves.

What a maths test on trigonometry looks like

A right-angled triangle solved with the basic relations, the laws of sines and cosines on any triangle, the area with the trigonometric formula, and almost always a problem with an unknown angle to discuss.

The mistake that costs the grade

Using the law of sines when the data are two sides and the angle between them: there you need the law of cosines. And, in the case of two sides and an angle that is not the included one, forgetting that the law of sines can give two triangles that are both valid.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.