Year 4 opens with the addition, double-angle and half-angle formulas and, right after, with goniometric equations. They are formulas that turn sines and cosines into other sines and cosines: if you don’t have in your head what they equal at the notable points and how they behave in the other quadrants, you are left with expressions you can neither simplify nor check, and every equation becomes one more unknown instead of a direct application. In physics the opening chapter is mechanical waves, and a wave is an oscillation that repeats in space — without the simple harmonic motion of Year 3 it remains a formula without an image.
The syllabus, block by block
The unit circle and notable values. Sine and cosine are not calculator keys: they are the two coordinates of a point turning on a circle of radius 1. Everything follows from there.
It is the squares that add up to 1, not sine and cosine on their own: for , , not .
Associated angles. Every goniometric equation in the second block of Year 4 ends with the same question — ”: and besides ?” — and associated angles are the answer. They don’t have to be learned one by one: they are read off the drawing.
The sine stays the same in the second quadrant (the ordinate is still positive), the cosine changes sign (the abscissa is negative) — it is the most common mistake in the chapter.
Simple harmonic motion. Mechanical waves, sound and then light are all oscillations: every point of the medium performs a simple harmonic motion, and the wave is the same oscillation repeated a little further on in time.
and differ by a factor of : they are different quantities, not two names for the same thing.
The mistakes that cost the most
The fundamental identity is squared, not to the first power: is false in general, it is that equals 1.
The cosine of supplementary angles changes sign, the sine doesn’t — it is the opposite of what intuition suggests from symmetry, and it is worth checking with a value: is negative, is not.
is not the frequency: with you don’t write but first calculate and only then — skipping the step gives a result more than six times too small, and in a test it doesn’t match any of the quantities asked for.
To review the unit circle and associated angles there is the Year 3 practice test on goniometry and the circle, and on motion in the plane — from which simple harmonic motion comes — the one on motion in the plane; on the Year 4 test that follows right after you’ll find the guide to goniometry and trigonometry. For exponentials and logarithms, which run alongside goniometry during Year 4, there is exponentials and logarithms in Year 4.
The Italian version of this post comes with a downloadable document that covers all nine topics in full: every formula, one worked example per topic, the typical mistake where needed and twenty-seven exercises with the complete step-by-step solution.
If you would rather have guided revision than go it alone, maths and physics tutoring start exactly from here — write to me and we’ll see where to concentrate the first lessons.
