Why Does the Cosine Addition Formula Have a Minus Sign? A Geometric Explanation Using Vector Rotation

The cosine addition formula is:

But why is there a minus sign between the two products? Why do we subtract sin A sin B rather than add it?

We can understand the reason geometrically by decomposing a vector into two perpendicular components, translating those components to the origin, and then rotating them through an angle B.

1. Decompose the Original Vector

Consider a unit vector OP making an angle A with the positive x-axis.

Its endpoint P has coordinates:

We can decompose this vector into two perpendicular components:

• Horizontal component: , pointing right.
• Vertical component: , pointing upward.

The original vector is the sum of these two components.

Notice that the vertical component is initially located at , extending from to . It does not initially lie along the y-axis.

2. Translate Both Components to the Origin

Here is an important geometric principle:

A vector can be translated parallel to itself without changing its magnitude or direction.

Using this principle, we can shift both components to the origin:

• The horizontal component becomes a vector of length along the positive x-axis.
• The vertical component becomes a vector of length along the positive y-axis.

These translated vectors represent exactly the same components as before.

This step is important because it allows us to rotate both component vectors about the origin through the same angle B.

3. Rotate the Horizontal Component

Consider the horizontal component, initially pointing right, with magnitude .

Rotate it counterclockwise through angle B.

It now points upward and rightward, remaining in the first quadrant.

Its horizontal projection, measured relative to the original fixed x-axis, is:

The projection is positive because it points towards the positive x-axis.

4. Rotate the Vertical Component

Now consider the vertical component, initially pointing upward, with magnitude .

Rotate it counterclockwise through the same angle B.

It now points upward and leftward, entering the second quadrant.

Its horizontal projection, measured relative to the same fixed x-axis, is:

The projection is negative because it points towards the negative x-axis.

Notice the important distinction: both components rotate counterclockwise through the same angle, but their different starting directions produce horizontal projections with opposite signs.

5. Combine the Horizontal Projections

The resulting horizontal coordinate is the sum of the two horizontal projections:

After rotation, the original unit vector makes an angle A + B with the positive x-axis.

Therefore, its horizontal coordinate is:

Equating the two expressions gives:

Conclusion: Where Does the Minus Sign Come From?

The minus sign is a consequence of the directions of the horizontal projections.

• The rotated horizontal component contributes a positive projection:

• The rotated vertical component contributes a negative projection:

Both projections are measured relative to the original, fixed coordinate axes.

The essential insight is that the minus sign does not arise simply because a component becomes smaller after rotation. It arises because the rotated vertical component contributes in the negative x-direction.

This geometric interpretation explains why the cosine addition formula involves subtraction rather than addition.