The cosine addition formula is

cos(A+B) = cos A cos B − sin A sin B

At first glance, the minus sign can seem mysterious.

Why is the contribution from cos A positive while the contribution from sin A is negative?

And if the original cos A component becomes smaller when it tilts, why don’t we subtract that decrease?

The clearest way to understand this is to focus on projections and direction.

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Q1. Where do cos A and sin A come from?

Consider a unit radius making an angle A with the horizontal axis.

Its Cartesian components are:

Horizontal component = cos A

Vertical component = sin A

So the radius can be represented by the coordinate pair:

(cos A, sin A)

At this stage:

• cos A points to the right.
• sin A points upward.

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Q2. What happens when the radius is rotated through B?

Now rotate the entire radius anticlockwise through an additional angle B.

The final angle is:

A+B

Our goal is to find the horizontal and vertical components of the new radius.

The important idea is:

Instead of trying to calculate the final components all at once, track what happens to the original cos A and sin A components separately.

We can then add their contributions.

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Q3. What happens to the original cos A component?

Originally, cos A is a horizontal segment:

O ─────────→
     cos A

Now rotate this segment anticlockwise through B.

For the simple diagram where B is less than 90°, the segment becomes tilted upward and to the right:

        ↗
       /
      /
     O

Because the segment is tilted, it now has:

• a horizontal projection
• a vertical projection

The horizontal projection points to the right.

Therefore, its horizontal contribution is positive.

Its length is:

cos A cos B

So the horizontal contribution is:

+cos A cos B

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Q4. Why does the horizontal component of cos A become smaller?

Originally, the entire length cos A was horizontal.

After tilting, only part of that length remains horizontal.

Therefore:

original horizontal length = cos A

new horizontal projection = cos A cos B

For an acute B, this is smaller than cos A.

So your intuition is correct:

The cos A component does lose some horizontal length when it tilts.

But this does not produce a negative sign.

Why?

Because the remaining horizontal projection still points to the right.

For example:

+0.8 → +0.6

The magnitude decreased, but the direction remained rightward.

Therefore it is still positive.

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Q5. Could we calculate the amount that cos A loses?

Yes.

The amount of horizontal length lost is:

cos A − cos A cos B

But the remaining horizontal component is:

cos A − (cos A − cos A cos B)

which simplifies to:

cos A cos B

So the decrease is real, but we don’t need to introduce it into the addition formula separately.

We directly calculate the remaining projection.

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Q6. What happens to the original sin A component?

Now consider the original vertical component:

      ↑
      |
      | sin A
      |
      O

Rotate it anticlockwise through B.

For the simple first-quadrant picture, it tilts toward the upper-left:

    ↖
   /
  /
 O

It now has:

• a horizontal projection pointing left
• a vertical projection pointing up

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Q7. Why is the horizontal contribution from sin A negative?

Look only at its horizontal projection.

It points:

←

Left is the negative horizontal direction.

Therefore its horizontal component is negative.

Its magnitude is:

sin A sin B

So its signed horizontal contribution is:

−sin A sin B

This is the source of the minus sign in the cosine addition formula.

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Q8. Are the plus and minus signs therefore based on whether the component gets smaller or larger?

No.

This is the most important distinction.

The sign is determined by direction.

For the rotated cos A component:

horizontal projection → right

Therefore:

+cos A cos B

For the rotated sin A component:

horizontal projection → left

Therefore:

−sin A sin B

The fact that the cos A component becomes smaller is a statement about magnitude.

The fact that the sin A component points left is a statement about direction.

These are different ideas.

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Q9. Why doesn’t the fact that sin A initially has no horizontal component determine its minus sign?

It doesn’t.

Initially, sin A has zero horizontal component.

After rotation, it develops a horizontal component.

But the fact that it started from zero does not tell us whether that new component is positive or negative.

We have to look at the direction.

After the anticlockwise rotation in our diagram, it points left.

Therefore:

negative horizontal component

The direction determines the sign.

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Q10. What happens if B becomes larger?

This is an important test of our reasoning.

Suppose B becomes greater than 90°.

The rotated cos A component can eventually point to the left:

B < 90°       B = 90°       B > 90°

    ↗             ↑             ↖
    |             |             |
    O             O             O

Its horizontal projection therefore changes:

right → zero → left

And the expression

cos A cos B

automatically reflects this.

For B less than 90°:

cos B > 0

so:

cos A cos B > 0

At B = 90°:

cos B = 0

so:

cos A cos B = 0

When cos B becomes negative:

cos A cos B < 0

Thus the same expression handles the change in direction.

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Q11. So does the trigonometric function determine the sign?

It is better to say this precisely.

The geometry determines what projection we need.

For the rotated cos A component, the horizontal projection has the form:

cos A cos B

For the rotated sin A component, the horizontal projection has the form:

−sin A sin B

Then the values of sin B and cos B automatically take their own signs as B moves through different quadrants.

So there are two things working together:

1. Geometry determines the projection and its orientation.
2. The signs of sin B and cos B automatically adapt the result for different values of B.

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Q12. Can we summarize the two rotated components?

Yes.

Original cos A component

After rotation:

       ↗
      /
     /
    O

Horizontal projection:

+cos A cos B

Vertical projection:

+cos A sin B

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Original sin A component

After rotation:

   ↖
  /
 /
O

Horizontal projection:

−sin A sin B

Vertical projection:

+sin A cos B

For this simple first-quadrant diagram, both vertical projections point upward.

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Q13. How do we get the cosine addition formula?

The final horizontal coordinate is obtained by adding the two horizontal contributions.

From the original cos A component:

+cos A cos B

From the original sin A component:

−sin A sin B

Therefore:

cos(A+B) = cos A cos B − sin A sin B

The minus sign appears because the two horizontal contributions point in opposite directions.

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Q14. How do we get the sine addition formula?

Now add the vertical contributions.

From the original sin A component:

+sin A cos B

From the original cos A component:

+cos A sin B

Therefore:

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

Both contributions point upward in the basic diagram, so they are added.

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Q15. What is the complete component picture?

The entire derivation can be summarized as:

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

Therefore:

Horizontal

cos A cos B − sin A sin B

= cos(A+B)

Vertical

sin A cos B + cos A sin B

= sin(A+B)

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Q16. What is the most important lesson?

There are three separate ideas that should not be confused.

1. Projection magnitude

Tilting cos A reduces its horizontal projection:

cos A → cos A cos B

2. Direction

The horizontal projection of the rotated cos A points right in the simple diagram:

+cos A cos B

The horizontal projection of the rotated sin A points left:

−sin A sin B

3. Changing B

If B becomes large enough to move a rotated component into another quadrant, the signs of sin B and cos B automatically change as appropriate.

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The Key Insight

A decrease in length does not create a minus sign. Direction creates the sign.

The original cos A component becomes less horizontal when it tilts, but its horizontal projection still points right in the basic diagram. Therefore:

+cos A cos B

The original sin A component develops a horizontal projection pointing left. Therefore:

−sin A sin B

Adding those two signed horizontal contributions gives:

cos(A+B) = cos A cos B − sin A sin B

This is the clean geometric reason for the minus sign.