The formula

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

can initially look like a formula that simply has to be memorized.

But there is a useful geometric technique that allows us to reconstruct the formula from components.

The key idea is:

Take the original horizontal and vertical components, use each component as the radius of an auxiliary circle, rotate that radius through angle B, and calculate the resulting vertical contribution.

This article focuses particularly on why the two contributions are different:

$\cos A\sin B$ $\sin A\cos B$

1. Start with a unit circle at angle A

Consider a unit circle with centre O.

Draw a radius to a point P making angle A with the horizontal.

Because the radius is 1, the point P has:

Horizontal component = cos A

Vertical component = sin A

So we can think of the original unit radius as having two components:

Horizontal component = cos A

Vertical component = sin A

The important quantities are therefore:

$\text{Horizontal component}=\cos A$ $\text{Vertical component}=\sin A$

2. Now increase the angle by B

Suppose we rotate the direction anticlockwise through an additional angle B.

The final angle becomes:

$A+B$

Our objective is to understand the final vertical component.

Instead of trying to calculate it all at once, we divide the problem into two parts.

We ask:

What does the original horizontal component contribute to the final vertical component?

and then:

What does the original vertical component contribute?

This is the component-based technique.

3. First contribution: start with cos A

The original horizontal component has length:

$\cos A$

We now use this length as the radius of an auxiliary circle.

So create a smaller circle centred at O with:

$\text{radius}=\cos A$

The purpose of this new circle is not to replace the original unit circle.

It is simply a convenient geometric tool for examining what happens to a component when the direction is changed by B.

4. Rotate the cos A radius through B

Take the radius of length cos A along the horizontal direction.

Now rotate it anticlockwise through angle B.

We obtain a right triangle.

Here:

$OF=\cos A$

and:

$FT=\text{vertical component}$

The angle between the horizontal direction and OF is B.

5. Why do we use sin B here?

Look at the triangle relative to angle B.

The sides are:

Hypotenuse = OF = cos A

Opposite = FT

Adjacent = OT

The vertical component FT is opposite angle B.

Therefore, using the definition of sine:

$\sin B=\text{opposite}/\text{hypotenuse}$

So:

$\sin B=FT/\cos A$

Multiply by cos A:

$FT=\cos A\sin B$

Therefore:

The original horizontal component contributes cos A sin B to the final vertical component.

This is where the term

$\cos A\sin B$

comes from.

6. Now consider the original vertical component

The original vertical component has length:

$\sin A$

We now use sin A as the radius of another auxiliary circle.

This time there is an important difference.

The original component is vertical, not horizontal.

So initially our radius points upward:

$\text{radius}=\sin A$

Now rotate this radius anticlockwise through B.

Because it starts vertically upward, it tilts toward the left.

7. The second right triangle

After the rotation, we can represent the situation as a right triangle.

The sloping side is the radius.

Therefore:

$\text{hypotenuse}=\sin A$

But notice something very important:

Angle B is measured from the vertical direction.

That changes which side is adjacent and which is opposite.

8. Why do we use cos B this time?

Look carefully at the angle B.

It is between:

the vertical side, and
the sloping radius.

The vertical component touches angle B.

Therefore, the vertical component is the adjacent side.

The sloping radius is the hypotenuse.

So:

$\text{adjacent}=\text{vertical component}$ $\text{hypotenuse}=\sin A$

Now use the definition of cosine:

$\cos B=\text{adjacent}/\text{hypotenuse}$

Therefore:

$\cos B=\text{vertical component}/\sin A$

Multiplying by sin A:

$\text{vertical component}=\sin A\cos B$

So:

The original vertical component contributes sin A cos B to the final vertical component.

9. Why one term uses sine and the other uses cosine

This is perhaps the most important lesson from the construction.

It is tempting to think:

“Sine means vertical and cosine means horizontal.”

That is not the correct rule.

Instead:

$\sin=\text{opposite}/\text{hypotenuse}$ $\cos=\text{adjacent}/\text{hypotenuse}$

The appropriate function depends on where the angle is located.

For the cos A component

The radius starts horizontally.

After rotation through B, the vertical component is opposite B.

Therefore we use:

$\sin B$

giving:

$\cos A\sin B$

For the sin A component

The radius starts vertically.

After rotation through B, the vertical component is adjacent to B.

Therefore we use:

$\cos B$

giving:

$\sin A\cos B$

10. Add the two vertical contributions

We now have two contributions to the final vertical component.

From the original horizontal component:

$\cos A\sin B$

From the original vertical component:

$\sin A\cos B$

Therefore the final vertical component is:

$\sin A\cos B+\cos A\sin B$

But after rotating the original direction through B, the final angle is:

$A+B$

The vertical component of a unit radius at angle A+B is:

$\sin(A+B)$

Therefore:

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

11. The technique behind the formula

The important thing is not merely remembering the final formula.

The technique can be summarized as follows.

Step 1 — Find the original components

For a unit radius at angle A:

$\text{Horizontal}=\cos A$ $\text{Vertical}=\sin A$

Step 2 — Treat each component separately

Use cos A as one auxiliary radius.

Use sin A as another auxiliary radius.

Step 3 — Rotate each through B

The new direction is obtained by an anticlockwise rotation through B.

Step 4 — Find the vertical component of each

For the horizontal component:

$\cos A\rightarrow\cos A\sin B$

For the vertical component:

$\sin A\rightarrow\sin A\cos B$

Step 5 — Add the vertical contributions

$\sin A\cos B+\cos A\sin B$

Step 6 — Recognize the final vertical coordinate

The final direction is A+B, so its vertical coordinate is:

$\sin(A+B)$

Therefore:

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

12. The deeper idea

The formula is therefore not just an isolated trigonometric identity.

It reflects a general idea about components.

A quantity pointing in one direction can be broken into components. When the direction changes, each original component contributes to the new directions.

In this case:

cos A contributes part of the final vertical component.

sin A also contributes part of the final vertical component.

The two contributions are:

$\cos A\sin B$

and:

$\sin A\cos B$

Their sum gives the final vertical component.

That is why the sine addition formula contains two terms.

Final takeaway

The most useful rule to remember is:

Do not associate sine permanently with vertical and cosine permanently with horizontal.

Instead, always ask:

1. Which angle am I using?

2. Which side is the hypotenuse?

3. Which side is opposite that angle?

4. Which side is adjacent to that angle?

Then:

$\text{opposite}/\text{hypotenuse}=\sin$ $\text{adjacent}/\text{hypotenuse}=\cos$

Using that simple technique, the terms

$\cos A\sin B$

and

$\sin A\cos B$

emerge naturally from the geometry rather than having to be memorized.

Why Does sin(A + B) Contain Both sin A cos B and cos A sin B?

1. Start with a unit circle at angle A

2. Now increase the angle by B

3. First contribution: start with cos A

4. Rotate the cos A radius through B

5. Why do we use sin B here?

6. Now consider the original vertical component

7. The second right triangle

8. Why do we use cos B this time?