Free · Year 3 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 3: the unit circle.

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

Programme covered by the test

Degrees and radians, conversions both ways
Sine, cosine and tangent on the unit circle
Standard values and sign in the four quadrants
The fundamental identity
Associated angles and reduction formulas
Graphs of sine, cosine and tangent: period and range
Translations, dilations, parity and symmetries of the graphs
Basic trigonometric equations in an interval

Two worked examples from the test

Question 1 · Goniometria: circonferenza goniometrica, angoli associati, grafici · Goniometria

Quanti radianti misura un angolo di $135°$ ?

A $\frac{4 π}{3}$
B $\frac{3 π}{4}$ correct answer
C $\frac{2 π}{3}$
D $\frac{3 π}{2}$

Solution

Per passare da gradi a radianti si moltiplica per $\frac{π}{180°}$ .

$135° \cdot \frac{π}{180°} = \frac{135 π}{180} = \frac{3 π}{4}$

Controllo di plausibilità: $135°$ sta fra $90° = \frac{π}{2}$ e $180° = π$ , e infatti $\frac{3 π}{4}$ è compreso fra i due.
L'opzione $\frac{4 π}{3}$ corrisponde a $240°$ : è la frazione capovolta, errore molto comune.

Question 2 · Goniometria: circonferenza goniometrica, angoli associati, grafici · Goniometria

Quanto vale $cos 240°$ ?

A $\frac{1}{2}$
B $- \frac{3}{2}$
C $\frac{3}{2}$
D $- \frac{1}{2}$ correct answer

Solution

$240°$ sta nel terzo quadrante, dove sia seno sia coseno sono negativi.
Angolo associato: $240° = 180° + 60°$ , e $cos (180° + α) = - cos α$ .
Quindi $cos 240° = - cos 60° = - \frac{1}{2} .$
L'opzione $- \frac{3}{2}$ è $cos 150°$ (oppure $cos 210°$ ): il valore $\frac{1}{2}$ appartiene a $60°$ , non a $30°$ , ed è questo lo scambio da evitare.

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 4 pages · 80 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What a maths test on basic trigonometric functions looks like

Degree-radian conversions both ways, standard values from memory, reduction to the first quadrant with associated angles, the fundamental identity used to find one function from another, and the graphs of sine, cosine and tangent with their transformations.

The mistake that costs the grade

Finding the cosine from the sine with the fundamental identity and writing only the positive root: the sign is decided by the quadrant, and must be discussed. The other constant mistake is solving a basic equation without staying within the interval the question asks for.

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.

Is the score not enough?

See which topics lost you the most marks, then read the trigonometry formulas you really need — or we look at your results together: the diagnostic is free.

Book the free diagnostic call The other Year 3 tests