Foundations — Trig functions for angles beyond 90° — ASTC rule (All, Sin, Tan, Cos)

This page builds every piece of notation the parent note leans on, starting from things a 12-year-old already knows: left, right, up, down. No symbol appears before we have drawn it.

0. What the little circle $°$ means — the degree

Before anything else, one tiny symbol shows up everywhere: $°$.

• $90°$ = a square corner (the two axes meet at $90°$).
• $360°$ = one complete lap.

Why the topic needs this: every angle in the parent note ($120°$, $210°$, $750°$, $-30°$) is written in degrees. If $°$ is undefined, none of those numbers mean anything. (Later, Radians and Degrees gives a second unit, the radian, where $\pi$ replaces $180°$.)

1. The $x$-axis, the $y$-axis, and a point $(x,y)$

Before angles, we need a map of the flat page.

• $x>0$ means right of centre, $x<0$ means left.
• $y>0$ means above centre, $y<0$ means below.

Figure 1 — The two axes and a point $P=(x,y)$. The blue segment shows the rightward distance $x$; the green dashed segment shows the upward distance $y$. Read every point as "(how far right, how far up)".

Why the topic needs this: the parent note redefines cosine and sine as $\cos \theta =x$ and $\sin \theta =y$. Those letters $x$ and $y$ are exactly the two numbers above. If you don't own "$(x,y)$ = (right, up)", the whole ASTC rule is unreadable.

2. The four quadrants

The two axes cut the page into four corners, which we number I, II, III, IV. (The order in which we count them — the spinning direction — is fixed carefully in §3; for now just note where each corner sits.)

Figure 2 — The four quadrants labelled with the signs of $x$ and $y$ in each. The grey curved arrow shows the counting order I → II → III → IV, which §3 names.

Why the topic needs this: ASTC is a statement about which quadrant your angle lands in. The signs of $x$ and $y$ shown above are the entire source of the "sign" half of the rule — nothing is memorised, it is read straight off this picture.

3. The angle $\theta$ and "anticlockwise from the $+x$-axis"

Now we spin.

• $\theta =0°$ points right along the $+x$-axis.
• $\theta =90°$ points straight up.
• $\theta =180°$ points left.
• $\theta =270°$ points straight down.

Negative angles just mean turn the other way (clockwise): $-30°$ is a small clockwise nudge, landing in QIV.

Angles past $360°$ mean you completed a full turn and kept going: $750°$ is two full turns ($720°$) plus $30°$ more.

Why the topic needs this: "$120°$ is in QII" only makes sense once "anticlockwise from the right" is fixed. This convention is what lets one number $\theta$ point to one spot.

4. The unit circle and the point $P$

Figure 3 — The unit circle (blue). The orange terminal ray at angle $\theta$ meets the circle at $P$. The point's horizontal position is $\cos \theta$ and its vertical position is $\sin \theta$; the dashed green segment and grey base form the right triangle behind $\tan \theta =y/x$.

The little equation $x^2+y^2=1$ is just Pythagoras: the horizontal leg $x$, the vertical leg $y$, and the radius $1$ as the slanted side of a right triangle. See Unit Circle for the full construction.

Why radius 1? With radius $1$, the "adjacent over hypotenuse" of the old right-triangle definition becomes just "$x$ over $1$" $=x$, and "opposite over hypotenuse" becomes "$y$ over $1$" $=y$. Choosing radius $1$ deletes the division so the coordinates are the ratios.

5. Where cos, sin, tan came from — the right triangle

The words "cosine", "sine", "tangent" started life as ratios of triangle sides. We meet them here so the unit-circle version above feels earned, not magic.

Why the topic needs this: the parent says the unit-circle definition "agrees with the triangle definition for acute $\theta$". That sentence is only convincing if you have both pictures side by side — hence this section.

6. The reference angle ${\theta }^{\prime }$

Picture the terminal ray in any quadrant and swing to whichever side of the horizontal line is closest:

• QI ($0°<\theta <90°$): the ray is already close to the $+x$-axis, so ${\theta }^{\prime }=\theta$.
• QII ($90°<\theta <180°$): swing to the $-x$-axis (at $180°$): ${\theta }^{\prime }=180°-\theta$.
• QIII ($180°<\theta <270°$): swing to the $-x$-axis (at $180°$): ${\theta }^{\prime }=\theta -180°$.
• QIV ($270°<\theta <360°$): swing to the $+x$-axis (at $360°$): ${\theta }^{\prime }=360°-\theta$.

Why the topic needs this: ASTC gives only the sign. The size of every answer comes from $\sin {\theta }^{\prime }$, $\cos {\theta }^{\prime }$ or $\tan {\theta }^{\prime }$ of this friendly acute angle. See Reference Angles for the full quadrant walk.

7. The $\pm$ symbol and "sign vs magnitude"

The whole ASTC recipe is one clean split: $\text{(ratio of }\theta )=\underset{\text{sign: from quadrant}}{\underbrace{(\pm )}}\times \underset{\text{magnitude: from reference angle}}{\underbrace{(\text{ratio of }{\theta }^{\prime })}}.$

Why the topic needs this: the classic error (parent's "Mistake 2") is thinking ASTC gives the magnitude. Owning the sign/magnitude distinction inoculates you against it.

How these foundations feed the topic

The diagram below is a prerequisite map: read it top to bottom, following each arrow as "this idea is needed to build the next". Boxes near the top are the raw ingredients (degrees, axes); the single box at the bottom is the ASTC recipe everything flows into.

Equipment checklist

Each line below is a self-test: cover everything to the right of the :::, say your answer out loud, then uncover to check. If any one trips you, reread its section before starting the parent topic.

Connections

• Unit Circle — the circle these definitions live on.
• Reference Angles — the acute angle ${\theta }^{\prime }$ built in §6.
• Radians and Degrees — same angles measured with $\pi$ instead of $180°$.
• Trigonometric Identities — where $x^2+y^2=1$ becomes ${\sin }^2+{\cos }^2=1$.
• Graphs of Sin Cos Tan — the per-quadrant signs plotted as a wave.