The angle-addition formulas become much easier to understand when we look at the same geometry in three different views.

The three views are:

1. The actual final vector at A + B
2. The two original components after being rotated through B
3. The horizontal and vertical contributions recombined to reproduce the same final vector

The important idea is that these are not three different mathematical objects. They are three ways of looking at the same result.

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View 1: The final vector at A + B

Start with a unit circle centred at O.

Suppose the original radius makes an angle A with the positive x-axis.

Now rotate that radius anticlockwise through an additional angle B.

The final angle is therefore:

A + B

The final radius reaches a point P on the unit circle.

Because the radius has length 1, the coordinates of P are:

horizontal coordinate = cos(A + B)

vertical coordinate = sin(A + B)

So:

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

We can show its Cartesian components by dropping perpendiculars from P to the axes.

                 P
                /|
               / |
              /  | sin(A+B)
             /   |
            /    |
           O─────H
             cos(A+B)

This is the actual final vector.

There is no question here about whether the components are perpendicular: the horizontal and vertical projections are perpendicular because they are defined relative to the Cartesian axes.

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View 2: Break the original radius into its two components

Now we take a different approach.

Instead of looking only at the final vector, go back to the original radius at angle A.

Its two components are:

horizontal component = cos A

vertical component = sin A

So conceptually:

              ↑ sin A
              |
              |
              O────────→ cos A

These two components together make the original radius.

Now rotate the entire radius through angle B.

But instead of treating the radius as one object, we examine what happens to its two components separately.

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The original horizontal component: cos A

The horizontal component has length:

cos A

When we rotate it anticlockwise through B, it becomes a sloping segment.

It now has a horizontal contribution pointing to the right.

That contribution is:

cos A cos B

So:

cos A → cos A cos B horizontally

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The original vertical component: sin A

The original vertical component has length:

sin A

Initially it points straight upward.

After rotating it anticlockwise through B, it tilts toward the left.

Therefore, it develops a horizontal contribution pointing to the negative x-direction.

Its magnitude is:

sin A sin B

Because it points left, its signed horizontal contribution is:

−sin A sin B

So:

sin A → −sin A sin B horizontally

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An Important Point About View 2

The two rotated components in this construction do not have to be perpendicular to each other.

This is important.

Originally, cos A and sin A are perpendicular because they are horizontal and vertical Cartesian components.

But after we rotate both through B, they are both tilted.

Therefore, when we draw them separately after rotation, they are generally not perpendicular.

That does not cause a problem.

We are not claiming that these two tilted segments are the new Cartesian axes.

They are simply the rotated contributions of the original components.

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View 3: Put the Contributions Back Together

Now comes the most important step.

We take the horizontal contributions produced by the two rotated components.

From the original horizontal component:

+cos A cos B

From the original vertical component:

−sin A sin B

Because one points right and the other points left, their net horizontal contribution is:

cos A cos B − sin A sin B

This gives us the horizontal coordinate of the final point P.

But View 1 already told us what that horizontal coordinate is.

It is:

cos(A + B)

Therefore:

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

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The Same Process Happens Vertically

We can do exactly the same thing for the vertical coordinate.

The original vertical component, sin A, after rotation through B, contributes an upward amount:

sin A cos B

The original horizontal component, cos A, after rotation through B, contributes an additional upward amount:

cos A sin B

Therefore the final vertical coordinate is:

sin A cos B + cos A sin B

But from View 1, the final vertical coordinate is:

sin(A + B)

Therefore:

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

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The Three Views Together

We can now understand the entire technique as three stages.

View 1 — Actual final vector

The radius is at angle A + B.

Its Cartesian coordinates are:

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

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View 2 — Rotate the original components

Start with:

cos A horizontally

and

sin A vertically.

Rotate both through B.

The resulting tilted components are generally not perpendicular.

We calculate how much each contributes horizontally and vertically.

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View 3 — Reconstruct the final Cartesian components

Add the contributions in each direction.

Horizontal:

cos A cos B − sin A sin B

Vertical:

sin A cos B + cos A sin B

These reproduce the coordinates of the actual final point:

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

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Why This Technique Is Useful

The formulas can initially appear to be arbitrary combinations of sine and cosine:

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

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

But the three-view method reveals where every term comes from.

Each term represents a component contribution.

For cosine:

• cos A contributes horizontally through cos B
• sin A contributes horizontally through sin B, but toward the left

For sine:

• sin A contributes vertically through cos B
• cos A contributes vertically through sin B

So the formulas are not simply rules to memorize.

They describe how the original horizontal and vertical components change their directions when the entire radius is rotated.

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The Main Geometric Insight

The most important idea is this:

The final vector at A + B is one actual vector on one unit circle. The component method is a way of reconstructing its horizontal and vertical coordinates by following what happens to the original cos A and sin A components when they are rotated through B.

Therefore, the three views are equivalent:

View 1:
Actual vector at A + B

↓

View 2:
Original components rotated through B

↓

View 3:
Their horizontal and vertical contributions combined

↓

Same final Cartesian coordinates

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

This is why the component approach works: it reconstructs the same point that we can directly locate on the unit circle at angle A + B.

Why Is View 2 Necessary?

View 2 provides the bridge between A and A + B.

The final radius is at angle A + B, but rather than simply accepting its final horizontal and vertical components, we want to understand how they arise from the original radius at A.

We therefore start with the two original components:

• cos A — the original horizontal component
• sin A — the original vertical component

We then consider what happens to each component when the entire radius is rotated through B.

This allows us to express the final components in terms of both A and B.

For example, horizontally:

• The original cos A produces a rightward contribution of cos A cos B.
• The original sin A produces a leftward contribution of sin A sin B.

Therefore:

final horizontal component = cos A cos B − sin A sin B

But the final radius is at angle A + B, so this same horizontal component is cos(A + B).

Thus, View 2 is the bridge that connects the original angle A and the rotation B to the final angle A + B.

We are not expressing A and B in terms of each other. We are expressing the components of the final radius in terms of the original angle A and the additional rotation B.