Foundations — Trig ratios of standard angles — 0°, 30°, 45°, 60°, 90° (derive, don't memorize blindly)

This page assumes nothing. We build every piece the parent note leans on, one at a time, each resting on the one before. Notice the opening idea above uses only plain words — no special symbols — on purpose: we define each symbol before we ever use it.

1. The symbol $\theta$ — a name for an angle

Before we draw anything, we need one nickname.

A degree ($°$) is the unit for measuring "how much you turned". A full spin around a point is $360°$; a quarter of that spin is $90°$. When the parent note writes $\sin 30°$, it has simply replaced $\theta$ with the number $30°$.

2. The right angle and the right triangle

Look at the figure. The square-marked corner is the right angle. The three sides get names based on where you stand, that is, which corner your angle $\theta$ sits in:

• The hypotenuse — the longest side, always opposite the right angle. (Same for everyone.)
• The opposite — the side facing away from the angle $\theta$ we care about.
• The adjacent — the side next to $\theta$ (that isn't the hypotenuse).

3. What "ratio" means — the heart of everything

4. Sine, cosine, tangent — the three ratios (for acute angles)

Now we can earn the three symbols the parent note uses constantly.

• $\sin \theta$ ("sine") = how tall the triangle is, compared to its slope side.
• $\cos \theta$ ("cosine") = how wide the triangle is, compared to its slope side.
• $\tan \theta$ ("tangent") = how tall compared to how wide — the steepness.

5. The square-root symbol  ​

6. The Pythagorean theorem — where the roots are born

Before we use this, we need to know why the special triangles have the leg lengths they do — the parent note assumes them, but we will earn them. First, one shape we will lean on:

Now apply the theorem to each:

• 45° triangle (equal legs $1,1$): $c^2=1^2+1^2=2\Rightarrow c=\sqrt{2}$.
• 30-60-90 triangle (hyp $2$, short leg $1$): $b^2+1^2=2^2\Rightarrow b^2=3\Rightarrow b=\sqrt{3}$.

7. Rationalizing — turning $\frac{1}{\sqrt{2}}$ into $\frac{\sqrt{2}}{2}$

Why bother? Cleaner adding: $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}=\sqrt{2}$. More in Rationalization.

8. Complementary angles — the reason $\sin$ and $\cos$ mirror

First, one fact we will lean on:

9. The unit circle — extending to 0° and 90°

At $0°$ and $90°$ the triangle collapses (a side becomes length $0$), and — as we warned in §4 — a right triangle can't even hold those angles. The Unit Circle fixes this by giving the same ratios without needing a triangle.

Now read the two endpoints straight off the circle:

• At $\theta =0°$: the point sits at $(1,0)$ → $\cos 0°=1$, $\sin 0°=0$, and $\tan 0°=\frac{\sin 0°}{\cos 0°}=\frac{0}{1}=0$.
• At $\theta =90°$: the point sits at $(0,1)$ → $\cos 90°=0$, $\sin 90°=1$, and $\tan 90°=\frac{\sin 90°}{\cos 90°}=\frac{1}{0}$ is undefined (dividing by zero).

Notice the two endpoints are mirror opposites: at $0°$ everything is flat ($\tan =0$), at $90°$ everything is vertical ($\tan$ blows up).

How these foundations stack up

Read this ladder from the bottom up — each rung only needs the rung below it:

1. Right angle + right triangle (§2) give us the three named sides.
2. The ratio idea (§3) — size doesn't matter — makes those side-comparisons depend on the angle alone.
3. Together they define sine, cosine, tangent for acute angles (§4).
4. Separately, the square root (§5) plus the Pythagorean theorem (§6, using the equilateral triangle) turn the two special triangles into the exact lengths $\sqrt{2}$ and $\sqrt{3}$.
5. Feeding those lengths into the trig ratios, then tidying with rationalizing (§7), complementary angles (§8), and the unit circle (§9, which reaches the $0°$/$90°$ endpoints), produces every entry in the parent topic's standard-angle table.

Equipment checklist

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