Foundations — Double angle formulas — sin 2A, cos 2A (three forms), tan 2A

This page is the ground floor. It assumes you have seen none of the notation. Every symbol the parent note leans on is unpacked here, in the order that each one needs the previous.

1. What is an angle, and what is $A$?

The letter $A$ is just a name for an amount of turning. Picture a clock hand pinned at the centre. Start it pointing right (east). Rotate it anticlockwise. How far you rotated is the angle.

• Measured in degrees: a full turn is $360^{\circ }$, a quarter turn (straight up) is $90^{\circ }$.
• The little raised circle ${}^{\circ }$ just means "degrees", the way a raised dot means "kilograms" is written kg — it is a unit tag, not a mathematical operation.

Look at the figure: the blue hand is at angle $A$, the yellow hand is at $2A$ — exactly double the sweep. That single picture is what the whole topic is about: relating what happens at $A$ to what happens at $2A$.

2. The right triangle — where $\sin$, $\cos$, $\tan$ are born

Drop the rotating hand and look at a right-angled triangle (one $90^{\circ }$ corner, shown by a small square).

Name the sides relative to the angle $A$:

• Opposite — the side across the room from $A$.
• Adjacent — the side lying alongside $A$ (touching it, not the long slanted one).
• Hypotenuse — the longest side, always facing the right angle.

Why ratios and not lengths? If you enlarge the triangle (photocopy it bigger), every side grows but the ratio of any two sides stays fixed. So these ratios depend only on the angle $A$, never on the size of the triangle. That is exactly why they are useful "codes" for an angle.

Why the topic needs these. The parent note's formulas like $\sin 2A=2\sin A\cos A$ are relationships between these ratios. Without knowing what $\sin$ and $\cos$ mean, the formula is just squiggles.

3. The unit circle — angles beyond the triangle

A triangle only has angles between $0^{\circ }$ and $90^{\circ }$. But $2A$ can easily be more than $90^{\circ }$ (if $A>45^{\circ }$). So we need a picture that handles any angle, including obtuse and negative ones. That picture is the unit circle: a circle of radius $1$ centred at the origin.

This matches the triangle definition (radius $1$ = hypotenuse, so opposite$/1=\sin A$), but now it keeps working past $90^{\circ }$.

Why the topic needs it — signs. When $A$ is in the second quadrant, the $x$-coordinate goes negative, so $\cos A<0$. That is the whole reason the parent's Example 1 warns "acute ⇒ positive root". The circle is where signs come from.

4. Squares of ratios: the notation ${\sin }^{2}A$

The parent writes ${\cos }^{2}A$, ${\sin }^{2}A$, ${\tan }^{2}A$. This is a shorthand:

Why the topic needs it. Every $\cos 2A$ form (${\cos }^{2}A-{\sin }^{2}A$, $2{\cos }^{2}A-1$, $1-2{\sin }^{2}A$) is written with these squares.

5. The Pythagorean identity — the swap engine

Go back to the unit circle: the point $(\cos A,\sin A)$ sits on a circle of radius $1$. Pythagoras on that little triangle (horizontal leg $\cos A$, vertical leg $\sin A$, hypotenuse $1$) gives:

Why the topic needs it — this is why $\cos 2A$ has three forms. Rearranged, it says ${\sin }^{2}A=1-{\cos }^{2}A$ and ${\cos }^{2}A=1-{\sin }^{2}A$. So any ${\sin }^2$ can be traded for a ${\cos }^2$ and vice versa. That single "swap" is what turns one $\cos 2A$ formula into three. See Pythagorean Identity for its own deep dive.

6. The equals sign for identities, and "undefined"

Two more pieces of notation the parent uses silently:

• Identity ($=$ that holds for every $A$): $\sin 2A=2\sin A\cos A$ is true for all angles, not just one special value. It is a permanent rule, like $x+x=2x$.
• Undefined: $\tan A=\frac{\sin A}{\cos A}$, and dividing by zero is forbidden. When $\cos A=0$ (at $90^{\circ }$), $\tan A$ has no value — its picture is a vertical slant with infinite steepness. That is why the parent says $\tan 2A$ "blows up" when $2A=90^{\circ }$.

The figure shows the slant getting steeper and steeper as the angle approaches $90^{\circ }$: $\tan$ shoots up without bound. There is no finite number to write, so we call it undefined.

7. The addition formulas — the seeds

The parent assumes these. They are the true starting point (proved in Compound (Addition) Angle Formulas):

Here $X$ and $Y$ are just two angle-names, like $A$. The double-angle trick is: set both equal to $A$. That is the entire manoeuvre — no new machinery, just a substitution.

8. How it all feeds the topic

Read it top-down: an angle gives a triangle, the triangle gives the ratios, the circle fixes their signs, the identity powers the three $\cos 2A$ forms, and the addition formulas — fed by all of this — become the double-angle formulas the moment you set $X=Y=A$.

Equipment checklist

Test yourself — if any answer is fuzzy, reread that section before the parent note.