Foundations — Solving trig equations — general solutions, solutions in given range

Before you can read the parent note, you must be fluent in every symbol it throws at you. Below, each symbol gets three things: what it means in plain words, the picture it stands for, and why the topic can't work without it. They are ordered so each one leans on the ones above it.

1. An angle $\theta$ — and what "measuring" it means

The picture: imagine standing at the centre of a circle and pointing outward. Start pointing right (that's $\theta =0$) and swing your arm anticlockwise. How far you've swung is $\theta$.

Why the topic needs it: the whole subject is "for which angles $\theta$ is some statement true?" You cannot ask that question without a name for the unknown angle. $\theta$ is that name.

2. Two ways to measure a turn: degrees and radians

The picture: wrap a piece of string of length $1$ around the edge of a unit circle (radius $1$). The angle it covers is exactly one radian. It takes $2\pi \approx 6.28$ such strings to go all the way round.

Why the topic needs it: the parent note freely mixes $360^{\circ }$ and $2\pi$, $180^{\circ }$ and $\pi$. They are the same turns in two languages. If you can't switch between them instantly you'll misread every formula. See Unit Circle and Radian Measure for the full build.

3. The unit circle — the master picture

The picture: the point where angle $\theta$ lands has coordinates $(\cos \theta ,\sin \theta )$. That single sentence is the definition of sine and cosine — read on.

Why the topic needs it: every "why $\pm$?", "why $\pi -\alpha$?", "why does it repeat?" answer lives on this circle. It is the map you read solutions off.

4. Cosine and sine — the coordinates of the landing point

The picture: drop a straight line from the landing point down to the $x$-axis. The along-the-floor distance is $\cos \theta$; the height is $\sin \theta$. Both live between $-1$ and $+1$ because the point never leaves a radius-$1$ circle.

Why the topic needs it: solving $\sin \theta =\frac{1}{2}$ literally means "which angles put the height at $\frac{1}{2}$?" — a question about this coordinate.

5. Tangent — steepness, and why it repeats twice as fast

The picture: the line from the centre through the landing point has a steepness. Going straight right is slope $0$; going up-and-right at $45^{\circ }$ is slope $1$; going straight up the slope blows up to infinity (that's why $\tan 90^{\circ }$ is undefined — $\cos 90^{\circ }=0$, division by zero).

Why the topic needs it: the parent's tan general solution $\theta =n\pi +\alpha$ has period $\pi$ for exactly this reason.

6. Periodicity and period $T$ — "it repeats"

The picture: on a graph, the curve is a stamp that reprints itself over and over left to right. See Graphs of Trigonometric Functions.

Why the topic needs it: "add any whole number of periods to a solution and it's still a solution" — this is the engine that turns one answer into infinitely many.

7. Integers $n$ and the symbol $\mathbb{Z}$

The picture: a number line with dots only on the whole marks. Each dot is "which loop of the Ferris wheel" you're counting.

Why the topic needs it: the general solution is a machine with a dial. Set the dial $n$ to $0,1,-1,2,\dots$ and each setting spits out one more solution. $\mathbb{Z}$ says the dial can be any whole number.

8. The $\pm$ sign and $(-1{)}^n$ — reading the shortcuts

Why the topic needs it: these are the abbreviations inside the general-solution formulas. Miss them and the formulas look like nonsense.

9. Inverse trig — the "which angle?" button

The picture: if $\sin$ takes an angle and gives a height, ${\sin }^{-1}$ takes a height and gives back one angle — the principal one. Because many angles share a height, the calculator can only hand you one; the general solution recovers the rest. See Inverse Trigonometric Functions.

Why the topic needs it: every worked example starts by finding $\alpha ={\sin }^{-1}s$ (or cos/tan). $\alpha$ is the seed the general formula grows from.

10. Ranges and interval notation

The picture: a stretch of the number line with a filled dot at $0$ (included) and a hollow dot at $2\pi$ (excluded).

Why the topic needs it: step 4 of the recipe keeps only solutions inside the given range. You must read the boundary symbols correctly, or you'll wrongly keep or drop an endpoint.

Prerequisite map

Equipment checklist

Test yourself — reveal only after you've answered out loud.

Connections

• ↑ Parent: Solving trig equations
• Unit Circle and Radian Measure — where angles, radians and the $(\cos ,\sin )$ point come from.
• Graphs of Trigonometric Functions — periodicity seen as repeating waves.
• Inverse Trigonometric Functions — the principal value $\alpha$.
• Trigonometric Identities — used to simplify equations to $\sin /\cos /\tan =c$.
• Compound and Double Angle Formulae — needed for multiple-angle equations.