Goniometry in Year 4: the test

In Year 4 maths changes face. Up to Year 3 you worked almost always in the Cartesian plane — lines, parabolas, conics — and the tools were few and recognisable. In September of Year 4 the chapter on goniometric formulas opens (goniometria: the trigonometric functions and identities, on the unit circle) and in a few weeks the book goes from a handful of rules to a list that seems endless: addition, subtraction, double angle, half angle, parametric, sum-to-product, Werner. The first serious test usually arrives between the end of October and November, on goniometric equations; the December one is almost always on trigonometry with discussion, and in schools with a trimester plus pentamester it falls right before the mid-December class councils, that is the first official school report.

It is the moment when I always hear the same sentence: “I have to learn forty formulas”. It isn’t true, and above all it doesn’t work. In this article I tell you where the chapter really starts, where marks are lost in Year 4 tests and how to set up your revision — with three worked exercises of the kind you find in the test.

What the Year 4 test really asks

In the science high school (liceo scientifico) and the applied-sciences track the sequence is almost always this: from September to October the goniometric formulas and equations (elementary, linear, homogeneous) plus inequalities; from November to December trigonometry proper, that is right-angled triangles, the chord theorem, sines, Carnot, area and problems with an angular unknown to discuss. In the human sciences high school goniometry arrives later — usually between March and May, reduced to radians, associated angles and elementary equations — so there the test falls in spring and almost certainly doesn’t touch the parametric formulas.

The format is fairly stable: a few exercises in one or two hours, typically two equations, an inequality, an identity to prove and a problem on a triangle. An original idea is almost never required: what is required is to recognise the type of exercise and apply the right procedure without losing pieces along the way. And that is exactly where grades are lost, not in theoretical difficulty.

A note worth making straight away, because it comes back in June: at the science high school the failed subject in Year 4 is almost always “goniometry plus limits”. Not because they are the hardest topics of the year, but because the first builds up from October and the second arrives in May, when you are already tired. Whoever keeps goniometry under control in November usually doesn’t end up at the September catch-up exam; if you are already there, the practice test is Year 4 maths catch-up.

Six formulas, not forty

First a division of labour, so you don’t read the same things twice. The “static” part of the chapter — the unit circle, the notable values of $30°$ , $45°$ and $60°$ , associated angles, addition and double angle with their derivations — I have already written out in full in the post on the goniometry formulas you really need for the maturità (the maturità is Italy’s school-leaving exam): if those pieces aren’t yet automatic, start from there (and if the circle is still an abstract drawing, here you can move it by hand and watch sine and cosine change). Here I take them as given and go where Year 4 has its specific problems: half angle and parametric formulas, the three types of equation, trigonometry with discussion.

The core to make automatic is made of six formulas. The first four:

sin^{2} α + cos^{2} α = 1, tan α = \frac{sin α}{cos α}

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

The other two are the theorems that arrive new in Year 4. The law of sines, in any triangle with sides $a$ , $b$ , $c$ , opposite angles $α$ , $β$ , $γ$ and circumscribed circle of radius $R$ :

\frac{a}{sin α} = \frac{b}{sin β} = \frac{c}{sin γ} = 2 R

And the law of cosines, also called Carnot’s theorem:

a^{2} = b^{2} + c^{2} - 2 b c cos α

The reason it is worth stopping at six is practical, not philosophical. Goniometric formulas all look alike: under time pressure pure memory doesn’t tell $cos α cos β - sin α sin β$ from $cos α cos β + sin α sin β$ , and a wrong sign at the start wrecks the exercise even if the following steps are impeccable. And a list learned by heart decays: whoever learns forty lines in November remembers eight in June, when trigonometry comes back in solid geometry problems and, in Year 5, in function studies. Whoever knows how to derive remembers six and rebuilds the others on demand.

The other Year 4 formulas — associated angles, Werner, sum-to-product, plus the method for goniometric inequalities — are collected in full in the Year 4 formula sheet, with a worked example and a typical mistake for each section and twelve solved exercises: handy for the final revision, when you need everything to hand at a glance, not for learning them from scratch here. You’ll find it in Year 4: the whole maths syllabus in a formula sheet.

The formulas you can derive in thirty seconds

Half angle. Start from the form $cos 2 α = 1 - 2 sin^{2} α$ of the double angle and solve for $sin^{2} α$ : you get $sin^{2} α = \frac{1 - c o s 2 α}{2}$ , and setting $α \to \frac{α}{2}$

sin \frac{α}{2} = \pm \frac{1 - cos α}{2}, cos \frac{α}{2} = \pm \frac{1 + cos α}{2}

The sign is not chosen at random: it depends on the quadrant in which $\frac{α}{2}$ falls, and it is the first thing the teacher checks.

Chord theorem and area. From the law of sines it follows that a chord equals $2 R sin γ$ , where $γ$ is any inscribed angle that subtends it; and the area of a triangle is $\frac{1}{2} ab sin γ$ , because the height relative to $a$ is $b sin γ$ . They are two lines of derivation, not two formulas to memorise.

Parametric formulas. With $t = tan \frac{α}{2}$ it holds that:

sin α = \frac{2 t}{1 + t ^{2}}, cos α = \frac{1 - t ^{2}}{1 + t ^{2}}

This is the only pair worth keeping written on your mental formula sheet, but with a warning that is worth more than the formula itself: the parametric formulas always lose the solution $α = π + 2 k π$ , because there $tan \frac{α}{2}$ doesn’t exist. If you use the parametric formulas in an equation, you must check separately whether $x = π + 2 k π$ is a solution. Whoever doesn’t hands in an incomplete exercise.

Goniometric equations: where solutions get lost

The equations in the Year 4 syllabus are of three types, and each has its trap.

Elementary ( $sin x = a$ , $cos x = a$ , $tan x = a$ ). I have written all the families of solutions in the post on the maturità; here only the operational warning is needed, namely that for sine and cosine the families are two and not one. If $sin x = \frac{1}{2}$ , besides $x = \frac{π}{6} + 2 k π$ there is also $x = \frac{5 π}{6} + 2 k π$ , because on the circle there are two points with the same ordinate. Only the tangent has just one, with period $π$ .

Linear ( $a sin x + b cos x = c$ ). They are solved with the auxiliary angle method — factoring out $a^{2} + b^{2}$ and recognising an addition formula — or with the parametric formulas, with the check of $x = π + 2 k π$ mentioned above. The auxiliary angle method is faster and loses nothing: when you can, use that.

Homogeneous ( $a sin^{2} x + b sin x cos x + c cos^{2} x = 0$ ). You divide by $cos^{2} x$ to get a quadratic equation in $tan x$ . But dividing by $cos^{2} x$ is allowed only if $cos x \neq = 0$ : before dividing you must check whether $x = \frac{π}{2} + k π$ is a solution of the starting equation. Nine times out of ten it isn’t and the check costs one line; the tenth time you have saved half the exercise.

There is then a mistake that doesn’t depend on the type of equation but on the argument: if the unknown appears as $2 x$ or $x + \frac{π}{3}$ , the periodicity must be carried back to the real unknown. From $2 x = \frac{π}{3} + 2 k π$ it follows $x = \frac{π}{6} + k π$ , not $x = \frac{π}{6} + 2 k π$ : dividing by two also halves the period. It is perhaps the most frequent mistake of all in November tests.

Three worked exercises

1. Homogeneous equation

sin^{2} x - 3 sin x cos x + 2 cos^{2} x = 0

First the check: if $cos x = 0$ then $sin^{2} x = 1 \neq = 0$ , so $x = \frac{π}{2} + k π$ is not a solution and we can divide by $cos^{2} x$ :

tan^{2} x - 3 tan x + 2 = 0

Setting $t = tan x$ we have $t^{2} - 3 t + 2 = 0$ , that is $t = 1$ or $t = 2$ . So

x = \frac{π}{4} + k π or x = arctan 2 + k π, k \in Z

The period is $π$ and not $2 π$ because the equation has been reduced to the tangent.

2. Linear equation

sin x + 3 cos x = 1

Here $a^{2} + b^{2} = 1 + 3 = 2$ . Divide everything by 2:

\frac{1}{2} sin x + \frac{3}{2} cos x = \frac{1}{2}

The two coefficients are $cos \frac{π}{3}$ and $sin \frac{π}{3}$ , so the left-hand side is the addition formula for the sine:

sin (x + \frac{π}{3}) = \frac{1}{2}

From here the two families: $x + \frac{π}{3} = \frac{π}{6} + 2 k π$ or $x + \frac{π}{3} = \frac{5 π}{6} + 2 k π$ , that is

x = - \frac{π}{6} + 2 k π or x = \frac{π}{2} + 2 k π

A quick check, always worth doing: for $x = \frac{π}{2}$ you get $1 + 0 = 1$ , and for $x = - \frac{π}{6}$

- \frac{1}{2} + 3 \cdot \frac{3}{2} = - \frac{1}{2} + \frac{3}{2} = 1

Both correct.

3. Any triangle

In a triangle $A B C$ it is known that $a = 7$ , $b = 5$ and the included angle $γ = 60°$ . Calculate the third side and the angle $α$ .

The third side with Carnot:

c^{2} = a^{2} + b^{2} - 2 ab cos γ = 49 + 25 - 2 \cdot 7 \cdot 5 \cdot \frac{1}{2} = 39

so $c = 39 \approx 6.24$ .

For the angle $α$ the temptation is to use the law of sines, but that is exactly where the “discussion” that loses marks arises: from $sin α$ you get back two supplementary angles, and you have to choose the right one. With the law of cosines instead the angle is determined, because the cosine is one-to-one between $0°$ and $180°$ :

cos α = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c} = \frac{25 + 39 - 49}{2 \cdot 5 \cdot 39} \approx 0.240

hence $α \approx 76°$ . The consistency check is immediate: $a$ is the longest side, so $α$ must be the largest angle — and $76°$ is indeed larger than $γ = 60°$ and than the third angle, which is about $44°$ . Practical rule: find the sides with the law of sines, the angles with the law of cosines. It saves you almost all the discussions.

The mistakes I see most often in Year 4 tests

Dividing by $cos x$ (or by $sin x$ ) without checking that it isn’t zero. It is the number one way to lose half the solutions.
Forgetting the second family in equations with sine or cosine, or writing $+ 2 k π$ where $+ k π$ belongs.
Not carrying the period back to the unknown when the argument is $2 x$ , $3 x$ or $\frac{x}{2}$ .
Getting the quadrant wrong in the half angle. The $\pm$ in front of the root is not decorative: it has to be decided by looking at where $\frac{α}{2}$ falls.
Writing $sin^{2} x = sin x$ . It equals $∣ sin x ∣$ , and in identities to prove this detail is often the whole exercise.
Mixing degrees and radians in the same exercise: if you work in radians stay in radians to the end, and convert only if the text asks.
In trigonometry, not checking that the triangle exists. An angle found that, added to the others, exceeds $180°$ must be discarded: it is the discussion of the problem, not a formal detail.

The two weeks before the test

The revision that works is not rereading the chapter. It is this, in three steps.

Write the six formulas from memory on a blank sheet, every day, for a week. Two minutes a day. If one doesn’t come out, that is your priority.
Derive the other formulas from the six, without looking at the book: double angle, half angle, chord theorem, area. If it takes you more than a minute per formula, repeat the next day.
Do exercises divided by type, not at random. Five elementary equations in a row, then five linear, then five homogeneous. The test requires you to recognise the type in ten seconds, and recognition is trained only by seeing exercises close together, not alternated.

In the last two days, a timed paper in the format of the test, marked by classifying every mistake: missing theory, wrong calculation, or right procedure but solution lost. They are three different problems with three different remedies, and telling them apart is worth more than another ten exercises.

If the problem is not preparation but the blank that arrives in the first five minutes, it is worth working on that too: I wrote about it in how to manage test anxiety in maths and physics.

How we work on this in lessons

With goniometry my order of work is always the same: first the circle drawn, then the six formulas, then the derivations done aloud by the student — not by me. Whoever derives the double-angle formula on their own three times never forgets it; whoever reads it on the formula sheet thirty times confuses it anyway in December.

On trigonometry problems we work above all on the discussion: which solutions are geometrically acceptable, why an angle must be discarded, how to set up the condition on the angular unknown. It is a different skill from calculation and has to be trained separately. Lessons are online and one-to-one: you’ll find current rates and packages on the pricing page, useful for assessing the 10-hour package if Year 4 has got off to a bad start and a longer path is needed. The Year 4 syllabus as a whole, with the mistakes that cost the most, is in Year 4: the whole maths syllabus in a formula sheet.

The quickest way to see where you stand is to try the goniometry quiz: fifteen multiple-choice questions with no penalty, on notable angles, identities and formulas. If instead the test awaiting you is the December one, with triangles and discussion, the targeted paper is the trigonometry practice test: fifteen questions in fifty minutes on the chord theorem, the law of sines and Carnot’s theorem, with a breakdown of weak points by topic at the end. The correct answer to every question is free and needs no registration; the guided step-by-step solution unlocks with your first lesson. If after the quiz you see that equations hold up but trigonometry doesn’t (or vice versa), write to me for a free introductory call: we look at the syllabus of your test and I tell you honestly how many lessons you need. The full picture of how I set up the path is on the page for maths tutoring.

Goniometry and trigonometry in Year 4: the test