Trigonometry for the maturità: formulas

Trigonometry is one of the topics I see my students confuse most, and almost always for the same reason: the formulas are not hard, but there are many of them, they look alike, and anyone who learns them by heart without understanding where they come from will mix them up under stress — exactly when they matter most, as in the maturità (Italy’s school-leaving exam) or a timed class test.

In this post I line up only the formulas I really use with my students: the ones that keep coming back in exercises, not the full list from the textbook. And above all I explain how to avoid mixing them up, which is the real problem more than the list itself.

The starting point: the unit circle

Before the formulas, a basic review that is worth always keeping in mind: on the unit circle (radius 1, centre at the origin), for an angle $α$ measured anticlockwise from the $x$ -axis:

$cos α$ is the $x$ -coordinate of the point on the circle
$sin α$ is the $y$ -coordinate of the point on the circle
$tan α = \frac{sin α}{cos α}$ is the slope of the line joining the origin to the point (defined when $cos α \neq = 0$ )

This single mental picture — a point moving round a circle of radius 1 — is the key to not having to memorise the sign and value in every quadrant: if you know where the point is, you know the sign of sine and cosine without any tables. If you would rather watch it move than imagine it, the interactive unit circle lets you drag the point and shows sine, cosine, the exact values of the special angles and the two waves that come out of it.

Special angles and values

These are the values you should have ready without working them out each time, because they appear everywhere in exercises:

Angle	$0°$	$30°$	$45°$	$60°$	$90°$
Radians	$0$	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{π}{2}$
$sin$	$0$	$\frac{1}{2}$	$\frac{2}{2}$	$\frac{3}{2}$	$1$
$cos$	$1$	$\frac{3}{2}$	$\frac{2}{2}$	$\frac{1}{2}$	$0$
$tan$	$0$	$\frac{3}{3}$	$1$	$3$	undefined

A trick I always give to avoid mixing up the sine and cosine of $30°$ and $60°$ : notice that they mirror each other — $sin 30° = cos 60° = \frac{1}{2}$ and $sin 60° = cos 30° = \frac{3}{2}$ . It is no coincidence: $30°$ and $60°$ are complementary angles (they add up to $90°$ ), and for complementary angles sine and cosine always swap.

The fundamental identity and associated angles

The fundamental identity of trigonometry

sin^{2} α + cos^{2} α = 1

It comes straight from Pythagoras’ theorem applied to the triangle that sine and cosine form on the unit circle. It is by far the most used identity, both for finding $sin α$ from $cos α$ (or vice versa) and for simplifying complicated expressions.

Associated angles

Associated angles link the sine, cosine and tangent of an angle to those of a simpler “reference” angle. You do not need to learn them all by heart: three cases, rebuilt mentally from the circle diagram, are enough.

Supplementary angles ( $180° - α$ ): the point is reflected in the $y$ -axis.

sin (180° - α) = sin α cos (180° - α) = - cos α

Opposite angles ( $- α$ ): the point is reflected in the $x$ -axis.

sin (- α) = - sin α cos (- α) = cos α

Complementary angles ( $90° - α$ ): sine and cosine swap.

sin (90° - α) = cos α cos (90° - α) = sin α

The way not to mix them up under stress is not to memorise the six formulas: it is to draw the point on the circle, apply the symmetry required (horizontal flip, vertical flip, or swapping the axes) and read off the resulting sign. With a minute of practice it becomes faster than searching your memory for the formula.

Addition and double-angle formulas

These are the formulas I see confused most, because they look similar in form but have different signs for sine and cosine.

Addition and subtraction formulas

sin (α \pm β) = sin α cos β \pm cos α sin β

cos (α \pm β) = cos α cos β \mp sin α sin β

Note the detail that trips up almost everyone: in the sine formula the signs behave “normally” (the same sign on the left and on the right), whereas in the cosine formula they flip (the $+$ on the left gives $-$ on the right, and vice versa). A practical way to remember it: “cosine always does the opposite”.

Double-angle formulas

You get them by setting $β = α$ in the addition formulas — it is more useful to be able to derive them on the spot than to learn them as separate formulas, because that way you never lose them completely:

sin (2 α) = 2 sin α cos α

cos (2 α) = cos^{2} α - sin^{2} α = 2 cos^{2} α - 1 = 1 - 2 sin^{2} α

The three forms of the cosine of $2 α$ are equivalent (you move from one to another with the fundamental identity) and are used at different moments: the first when you know both the sine and the cosine of $α$ , the second and third when you know only one of the two and want to eliminate the other from the expression.

Elementary trigonometric equations

An equation is called elementary when it has the form $sin x = a$ , $cos x = a$ or $tan x = a$ , with a single unknown angle and no combinations.

For $sin x = a$ (with $- 1 \leq a \leq 1$ , otherwise there are no solutions): if $α$ is one particular solution, the general solution is

x = α + 2 k π or x = π - α + 2 k π, k \in Z

For $cos x = a$ (with $- 1 \leq a \leq 1$ ):

x = \pm α + 2 k π, k \in Z

For $tan x = a$ (defined for every real $a$ ):

x = α + k π, k \in Z

The difference to keep in mind: sine and cosine have period $2 π$ and give two families of solutions in the interval $[0, 2 π)$ (except in limiting cases such as $sin x = 1$ ), whereas tangent has period $π$ and gives only one family, because half a period later it already repeats the same value.

Trigonometric inequalities: the graphical method

For elementary inequalities — $sin x > a$ , $cos x \leq a$ and the like — the most reliable way, and the one I always teach first, is the graphical method on the circle:

Draw the unit circle.
Find the points where $sin x = a$ (or $cos x = a$ ), which divide the circle into arcs.
Identify visually on which arcs the required inequality holds (for $sin x > a$ : where the $y$ -coordinate is greater than $a$ ; for $cos x > a$ : where the $x$ -coordinate is greater than $a$ ).
Write the angular interval corresponding to the arc you found, adding $+ 2 k π$ for periodicity.

It is slower than the purely algebraic method in the first exercises, but it is almost impossible to get the direction of the inequality wrong if you work on the diagram — while it is very easy to get it wrong working only with symbols, especially when the inequality has a composite argument (e.g. $sin (2 x - \frac{π}{3})$ ).

The mistakes I see most often

Mixing up the sign in the cosine addition formula. It is mistake number one: the sign flips compared with the sine, and under stress people tend to apply the same pattern to both.
Forgetting the second family of solutions in equations with sine or cosine. Writing only $x = α + 2 k π$ and forgetting $x = π - α + 2 k π$ (or the equivalent for cosine) costs you half the solutions; it is not a detail.
Using degrees and radians in the same exercise without converting. If the exercise is set in radians and you substitute a value in degrees (or vice versa) without converting, the result is wrong even if the method is correct.
Applying associated-angle formulas without the diagram. Anyone who learns them as a list of six lines almost always mixes them up within a month; anyone who learns to rebuild them with the circle does not.
Simplifying $sin^{2} α + cos^{2} α$ as if it were $(sin α + cos α)^{2}$ . They are two different expressions: the first is always $1$ , the second expands as the square of a binomial and is not constant.

How it is used in the maturità

In the second paper of the maturità (the maths one), trigonometry rarely appears as a topic in its own right, but almost always as a tool inside another exercise: a function study with sine or cosine in the expression, a geometry problem that requires solving a trigonometric equation to find an angle, a limit or a derivative that simplifies only if you apply a double-angle formula correctly. That is why it is worth arriving at the exam with these formulas automatic, not recalculated from scratch each time: doing so slows an exercise down enormously, when without that slowdown it would be manageable in the time.

If you are reviewing function study and need a broader picture of the whole procedure, I have written a step-by-step guide: function study, the complete method. It is worth reading together with this post, because trigonometry often turns up right there as one of the ingredients, not as an isolated topic. You will find the full picture of how I work on these topics on the page dedicated to maths.

How we work on this in lessons

With trigonometry my approach is always the same: first the diagram on the circle, then the formula, never the other way round. Anyone who starts memorising formulas without having absorbed what sine, cosine and tangent represent on the circle systematically mixes them up within a few weeks, because they have no way of checking them “by eye” when memory wavers.

In lessons we work with exercises of increasing difficulty: first elementary equations and inequalities on their own, then the same inside an equation with a composite argument, then inside a wider exercise (function study, geometry problem) where trigonometry is only one step among many. It is in this last step that most students get stuck, because they have to recognise at a glance which formula is needed without anyone pointing it out — and that is exactly the skill we train.

Frequently asked questions

Do I have to learn all the trigonometry formulas by heart? The addition and double-angle formulas and the fundamental identity, yes: those must be known by heart because they are needed too often to derive each time. The associated angles, no: it is faster and more reliable to rebuild them with the diagram on the circle.

How do I remember the special values for $30°$ , $45°$ and $60°$ ? The most common trick is the pattern of increasing roots: the $sin$ of $0°, 30°, 45°, 60°, 90°$ is

\frac{0}{2}, \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}

which, once simplified, give exactly the values in the table. For cosine, the same sequence is read backwards.

Why does tangent have period $π$ and not $2 π$ like sine and cosine? Because tangent is defined as the ratio $\frac{s i n α}{c o s α}$ , and after half a turn (that is, $π$ ) sine and cosine both change sign: their ratio stays the same. Sine and cosine on their own, instead, change sign after half a turn and return to their original value only after a full turn.

Are the double-angle formulas useful outside trigonometry? Yes, they come up often in calculus (derivatives and integrals of trigonometric functions) and in analytic geometry. It is worth learning them well in high school because you meet them again, almost identical, in the first year of university.

How long does it take to be solid in trigonometry? With regular effort, 2-3 weeks of targeted exercises (not hours crammed into a weekend) are usually enough to go from “confused” to “solid” on the basic topics. Most of the time goes into repeated practice, not into understanding the theory, which in itself is short.

If you want to practise on these topics right away, there is the free trigonometry review quiz: 15 multiple-choice questions with no penalty, designed to consolidate what you have read here. And if after the quiz you find that some topics are still weak, the quickest route is a free 30-minute introductory call: we look together at where you stand and build a targeted plan, without going back over what you already know. For the class-test version for Year 4, with trigonometry problems and discussion, there is trigonometry in Year 4: preparing for the test; to fit it into the calendar of your maturità year, Year 5: the plan for the maturità from September.

Trigonometry for the maturità: the formulas you need