Half angle formulas — derivations from double angle: Foundations

Before you can derive a single half-angle formula, you must be fluent with a handful of symbols and pictures the parent note leans on without pausing. We build each one from nothing, in the order they depend on each other.

1. The angle — and what "half" of it means

An angle is an amount of turning. Picture a clock hand starting flat along the positive horizontal direction and rotating anticlockwise. How far it has swung is the angle.

We name angles with letters like $θ$ (Greek "theta") or $A$ . These are just labels for "some amount of turn".

Figure s01 shows this directly: the orange arrow is a full turn $A$ , and the blue arrow beside it is exactly half that turn, $A /2$ . Notice the blue wedge is half the width of the orange one.

Figure — Half angle formulas — derivations from double angle

Figure s01 — An angle $A$ (orange) and half of it $A /2$ (blue) on the unit circle. The blue wedge fills exactly half the orange wedge.

Why the topic needs this: the entire subject asks "given facts about $A$ , what about $A /2$ ?" You cannot even state the question without the idea of halving a turn — and, crucially (see §6), halving the turn can land you in a different region of the plane.

2. Two numbers born from an angle: $cos$ and $sin$

Draw a circle of radius $1$ centred at the origin — the unit circle. Swing a point around it by angle $θ$ . That point has a horizontal position and a vertical position.

In figure s02, the blue segment along the floor is $cos θ$ (how far right the point sits) and the green segment standing up is $sin θ$ (how high it sits). The red dot is the point itself, and the small grey wedge marks the angle $θ$ .

Figure s02 — $cos θ$ is the horizontal (blue) coordinate and $sin θ$ the vertical (green) coordinate of the red point on the unit circle.

Why "steepness" language works: as the point climbs, $sin θ$ grows; as it moves left, $cos θ$ shrinks. The angle controls these two numbers, so knowing $cos A$ pins down a lot about $A$ .

3. Squaring, and the notation $cos^{2} θ$

To square a number is to multiply it by itself: $3$ squared is $3 \times 3 = 9$ . Geometrically, squaring a length gives the area of a square with that side.

Why the topic needs this: every half-angle formula solves for a squared quantity first ( $sin^{2} \frac{A}{2}$ , $cos^{2} \frac{A}{2}$ ). The squares appear because the double-angle identity is built from products like $cos θ \cdot cos θ$ .

4. The seed identity: double-angle cosine

The whole topic starts from a single fact about $cos$ of a doubled angle. We state it here, because §5 onward — and the parent note — reshape it constantly.

Why the topic needs this: everything the parent does is this equation, rearranged. To rearrange it you must first be able to remove one of its two squared functions — which is exactly what the next tool, the Pythagorean identity, provides.

5. The Pythagorean identity — the swap rule

Because our point sits on a circle of radius $1$ , its horizontal and vertical coordinates always satisfy one unbreakable relationship.

Figure s03 makes this literal: the blue side is $cos θ$ , the green side is $sin θ$ , and the red hypotenuse is the radius $1$ . Pythagoras on this right triangle is the identity.

Figure s03 — The right triangle inside the unit circle: $cos^{2} θ$ (blue leg) $+$ $sin^{2} θ$ (green leg) $=$ $1^{2}$ (red hypotenuse).

Rearranged, it is a swap rule — a licence to trade one squared function for the other: $cos^{2} θ = 1 - sin^{2} θ and sin^{2} θ = 1 - cos^{2} θ .$

Why the topic needs this — and which tool, why: the parent note takes the seed $cos 2 θ = cos^{2} θ - sin^{2} θ$ and turns it into two one-function versions. The only tool that lets you erase one of the two functions is this swap rule. We reach for Pythagoras (not, say, a derivative) precisely because the question is "how do I remove $sin^{2}$ so only $cos^{2}$ remains?" — an algebraic substitution question, which this identity answers exactly.

6. $tan θ$ — the ratio of the two coordinates

Squaring the ratio just squares top and bottom: $tan^{2} θ = \frac{s i n ^{2} θ}{c o s ^{2} θ} .$

Why the topic needs this: the tangent half-angle formula is obtained by dividing the sine result by the cosine result. That division only makes sense once you know $tan = sin / cos$ . And the reason the parent prefers $tan$ for the Weierstrass substitution (Weierstrass substitution) is that this single ratio is enough to rebuild both coordinates.

7. The $\pm$ sign and the four quadrants

The plane splits into four quadrants by the horizontal and vertical axes, numbered anticlockwise I, II, III, IV. In each, the signs of the coordinates differ:

Quadrant	angle range	$cos$ (horizontal)	$sin$ (vertical)
I	$0^{\circ} - 9 0^{\circ}$	$+$	$+$
II	$9 0^{\circ} - 18 0^{\circ}$	$-$	$+$
III	$18 0^{\circ} - 27 0^{\circ}$	$-$	$-$
IV	$27 0^{\circ} - 36 0^{\circ}$	$+$	$-$

Figure s04 colour-codes the same table onto the plane: read off each quadrant's pair of signs by whether the point sits left/right (sign of $cos$ ) and up/down (sign of $sin$ ).

Figure s04 — Signs of $cos$ (horizontal) and $sin$ (vertical) in each of the four quadrants. Quadrant IV has $cos > 0$ , $sin < 0$ .

Prerequisite map

Each block feeds the next: coordinates give you the functions, the seed identity and Pythagoras let you reshape it into one-function forms, tangent lets you divide, and quadrants let you fix the sign — together they produce the half-angle formulas.

Equipment checklist

Self-test: can you answer each without peeking?

What does

A /2

mean, and why can its quadrant differ from

A

's?

Half as much turning as

A

; halving a turn can cross a quadrant boundary, so read

A /2

's quadrant separately.

On the unit circle, which coordinate is

cos θ

and which is

sin θ

cos θ

is the horizontal coordinate,

sin θ

the vertical.

Does

cos^{2} θ

mean

cos (θ^{2})

No — it means

(cos θ)^{2}

, i.e. compute

cos θ

first, then square it.

State the double-angle cosine seed identity.

cos 2 θ = cos^{2} θ - sin^{2} θ

State the Pythagorean identity and what it lets you do.

sin^{2} θ + cos^{2} θ = 1

; it lets you swap

sin^{2}

for

1 - cos^{2}

(or vice versa) to remove one function.

Write

tan θ

and

tan^{2} θ

as ratios.

tan θ = sin θ / cos θ

and

tan^{2} θ = sin^{2} θ / cos^{2} θ

Why does a

\pm

appear in the half-angle formulas?

Taking a square root loses the sign, so both signs are possible until the quadrant of

A /2

picks one.

In which quadrant is

cos

positive but

sin

negative?

Quadrant IV (

27 0^{\circ}

–

36 0^{\circ}

Connections

Double angle formulas — the seed identity these foundations prepare you to reverse.
Pythagorean identity — the swap rule of §5.
Weierstrass substitution — built on the tangent half-angle of §6.
Exact trig values — where the quadrant sign choices of §7 pay off.
Product-to-sum formulas — a sibling manipulation of the same coordinate ideas.