The cosine addition formula is

cos(A+B) = cos A cos B − sin A sin B

The minus sign can initially seem puzzling.

If the original cos A component tilts as we rotate the radius, its horizontal projection becomes smaller. Why doesn’t that decrease produce a minus sign?

And if B becomes large enough, the directions of the rotated components can change. So how do we know which signs belong in the formula?

A useful way to resolve this is to look at what happens during an infinitesimally small anticlockwise rotation.

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Q1. What are the original components of the radius?

Consider a unit radius at angle A.

Its Cartesian components are:

horizontal = cos A

vertical = sin A

So the radius can be represented by:

(cos A, sin A)

We now rotate this radius anticlockwise through an additional angle B.

The final angle is:

A+B

Our objective is to determine the new horizontal and vertical components.

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Q2. What happens if we rotate the horizontal direction by an infinitesimally small amount?

Start with a purely horizontal direction:

→

Now rotate it anticlockwise through a very small positive angle.

It immediately becomes:

↗

Two things have happened:

• it still has a component pointing right
• it has developed a component pointing up

Therefore, for a positive small B:

horizontal contribution → positive

vertical contribution → positive

As B increases, the lengths of these projections are represented by:

cos B

and

sin B

Therefore the rotated horizontal unit direction is represented by:

(cos B, sin B)

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Q3. What happens to the vertical direction under the same tiny rotation?

Now start with the vertical direction:

↑

Rotate it anticlockwise through the same tiny positive angle.

It immediately becomes:

↖

Now the situation is different.

The rotated direction has:

• a component pointing left
• a component pointing up

Therefore:

horizontal contribution → negative

vertical contribution → positive

The corresponding components are:

−sin B

and

cos B

So the rotated vertical unit direction is:

(−sin B, cos B)

This is where the fixed minus sign in the rotation structure comes from.

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Q4. Why is the infinitesimal rotation so useful?

Because when B becomes large, the diagram can become visually confusing.

For example, the rotated horizontal direction goes through:

→ → ↗ → ↑ → ↖ → ←

Its horizontal component therefore changes from positive to zero and eventually to negative.

Likewise, the rotated vertical direction goes through:

↑ → ↖ → ← → ↙ → ↓

Its horizontal component also changes direction.

Instead of creating a new formula every time the direction changes, we use:

(cos B, sin B)

for the rotated horizontal direction, and

(−sin B, cos B)

for the rotated vertical direction.

The values of sin B and cos B automatically change sign as B moves through different quadrants.

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Q5. Does the minus sign therefore depend on B?

This is an important distinction.

The fixed minus sign in

(−sin B, cos B)

does not arise because B happens to be acute.

It comes from the orientation of the coordinate directions under an anticlockwise rotation.

For a tiny positive rotation:

horizontal direction:   → becomes ↗
vertical direction:     ↑ becomes ↖

The vertical direction therefore acquires a leftward horizontal component.

That establishes the minus sign.

As B becomes larger, sin B and cos B themselves change sign and automatically describe the continuing rotation.

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Q6. What happens to the original cos A component?

The original horizontal component has length:

cos A

After rotation through B, multiply the rotated horizontal unit direction by cos A.

We get:

cos A(cos B, sin B)

Therefore its contributions are:

horizontal = cos A cos B

vertical = cos A sin B

For a small positive B, both are positive.

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Q7. What happens to the original sin A component?

The original vertical component has length:

sin A

Its rotated unit direction is:

(−sin B, cos B)

Therefore:

sin A(−sin B, cos B)

gives:

horizontal = −sin A sin B

vertical = sin A cos B

So the two original components produce four contributions:

Original component	Horizontal	Vertical
cos A	cos A cos B	cos A sin B
sin A	−sin A sin B	sin A cos B

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Q8. Why doesn’t the decrease in the horizontal cos A component produce a minus sign?

Because a decrease in magnitude is not the same as a negative direction.

Initially, the cos A component is completely horizontal.

After a small rotation, part of it has moved into the vertical direction.

Therefore its horizontal projection becomes smaller:

cos A → cos A cos B

But for a small positive B, that projection still points right.

So its signed horizontal contribution is:

+cos A cos B

The fact that its magnitude has decreased does not change its sign.

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Q9. What happens if B becomes greater than 90°?

Now consider the rotated cos A direction.

As B increases:

→ → ↗ → ↑ → ↖

Eventually its horizontal component points left.

At that point:

cos B < 0

Therefore:

cos A cos B < 0

The expression has automatically changed sign.

We do not manually replace the plus with a minus.

The value of cos B already reflects the new direction.

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Q10. What happens to the rotated sin A direction?

The rotated vertical direction follows:

↑ → ↖ → ← → ↙ → ↓

Its horizontal component is represented by:

−sin B

Again, the trigonometric function handles changes as B varies.

For example:

• if sin B > 0, then −sin B < 0
• if sin B = 0, then −sin B = 0
• if sin B < 0, then −sin B > 0

So the horizontal direction changes automatically as the rotation continues.

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Q11. Is the infinitesimal rotation therefore the “litmus test”?

Yes — for understanding the fixed orientation of the rotation.

Ask this simple question:

What happens when I rotate each original coordinate direction through an infinitesimally small positive angle?

For the horizontal direction:

→ becomes ↗

So its components initially have signs:

(+ , +)

For the vertical direction:

↑ becomes ↖

So its components initially have signs:

(− , +)

Therefore the rotation has the structure:

horizontal direction → (cos B, sin B)

vertical direction → (−sin B, cos B)

Once this structure is established, the trigonometric functions extend it to every value of B.

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Q12. How does this produce the addition formulas?

The original radius has components:

(cos A, sin A)

After rotation:

Contribution from cos A

cos A(cos B, sin B)

Contribution from sin A

sin A(−sin B, cos B)

Add them:

(cos A cos B − sin A sin B,
cos A sin B + sin A cos B)

But after rotating the original radius through B, its final angle is A+B.

Therefore its coordinates are:

(cos(A+B), sin(A+B))

Matching the coordinates gives:

cos(A+B) = cos A cos B − sin A sin B

and

sin(A+B) = sin A cos B + cos A sin B

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Q13. What is the deeper idea behind the derivation?

The important discovery is that we don’t actually need to memorize the signs by looking at one particular triangle.

Instead, we can understand the rotation itself.

A tiny anticlockwise rotation tells us how the two basic coordinate directions initially move:

→ becomes ↗

↑ becomes ↖

From this we obtain the rotation structure:

(1, 0) → (cos B, sin B)

(0, 1) → (−sin B, cos B)

Then any vector can be built from these two directions.

For the radius at A:

(cos A, sin A)

the rotation gives:

(cos(A+B), sin(A+B))

through exactly the component calculation above.

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The Key Insight

The infinitesimal rotation reveals the direction structure.

When the horizontal direction is rotated anticlockwise by a tiny amount, it moves upward and remains rightward:

→ becomes ↗

When the vertical direction is rotated anticlockwise by a tiny amount, it moves leftward and remains upward:

↑ becomes ↖

That initial directional behavior establishes the rotation structure:

(cos B, sin B)

and

(−sin B, cos B)

The signs of sin B and cos B then automatically change as B moves through different quadrants.

So the minus sign in

cos(A+B) = cos A cos B − sin A sin B

is not simply a consequence of one particular acute-angle diagram.

It reflects the geometry of an anticlockwise rotation of the coordinate directions.