Understanding sin(A + B) by Tracking the Change in Vertical Height

The formula

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

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

But the formula becomes much more understandable if we ask a different question:

What happens to the vertical component when a unit radius initially at angle A is rotated anticlockwise through an additional angle B?

The image above illustrates this process step by step.

The key idea is that we will start with the original vertical height, sin A, and then calculate exactly how that height changes.

1. Start at angle A

Consider a unit circle with centre O.

Take a radius OP making angle A with the horizontal axis.

Because the radius has length 1, its components are:

Horizontal component = cos A

Vertical component = sin A

So the point P has coordinates:

$(\cos A,\sin A)$

The important quantity for us is its vertical height:

$\sin A$

We can therefore think:

At angle A, our starting vertical height is sin A.

2. Now rotate the radius through B

Next, rotate the radius anticlockwise through an additional angle B.

The final angle is therefore:

$A+B$

The new point lies at:

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

But we don’t want to assume the formula for sin(A+B).

Instead, we want to discover the new vertical height.

So we start with what we already know:

starting vertical height = sin A

and ask:

How much does this vertical height change when we rotate through B?

This is the central idea of the whole derivation.

3. There are two effects during the rotation

The original radius has two components:

Horizontal = cos A

Vertical = sin A

When we rotate the radius, both components affect the final vertical height.

This is why we eventually get two terms.

Let’s examine them separately.

4. Effect of the original horizontal component

The original horizontal component has length:

$\cos A$

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

Now take this radius horizontally and rotate it anticlockwise through angle B.

We obtain a right triangle.

For this triangle:

Hypotenuse = cos A

Angle = B

Vertical side = the new vertical contribution

Because the vertical side is opposite B:

$\sin B=\frac{\text{vertical contribution}}{\cos A}$

Therefore:

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

So the original horizontal component creates an additional upward contribution of:

$\cos A\sin B$

This is the first part of the change.

5. Effect of the original vertical component

Now consider the original vertical component:

$\sin A$

This part is slightly different.

Initially it points vertically upward.

When the entire direction is rotated anticlockwise through B, this component also rotates.

So it no longer contributes its entire original length sin A vertically.

Its new vertical contribution becomes:

$\sin A\cos B$

Why cosine?

Because in the auxiliary triangle, the angle B is measured from the vertical direction.

Therefore the vertical side is adjacent to B.

Using:

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

we obtain:

$\text{new vertical contribution}=\sin A\cos B$

6. This is the part that is easy to miss

At angle A, the vertical component was:

$\sin A$

After rotation through B, that component contributes only:

$\sin A\cos B$

vertically.

Therefore, the original vertical component has lost some vertical height.

The amount lost is:

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

Factor out sin A:

$\sin A(1-\cos B)$

So there are actually two effects:

Gain in vertical height

From the original horizontal component:

$+\cos A\sin B$

Loss in vertical height

From the original vertical component:

$-\sin A(1-\cos B)$

This is the key to understanding why we cannot simply take:

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

7. Calculate the total change

The total change in vertical height is:

gain − loss

Therefore:

$\text{change in vertical height}=\cos A\sin B-\sin A(1-\cos B)$

Expand the second term:

$\text{change}=\cos A\sin B-\sin A+\sin A\cos B$

So the rotation changes the original vertical height by:

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

8. Start with sin A and add the change

Now we can follow exactly the intuitive approach:

Final height = original height + change in height

The original height was:

$\sin A$

Therefore:

$\text{final height}=\sin A+\left[\cos A\sin B-\sin A+\sin A\cos B\right]$

Now something beautiful happens.

The original:

$+\sin A$

and the loss:

$-\sin A$

cancel.

We are left with:

$\text{final height}=\cos A\sin B+\sin A\cos B$

9. But what is the final height?

After rotating the radius through B, its final angle is:

$A+B$

For a unit circle, the vertical coordinate of a point at angle A+B is:

$\sin(A+B)$

Therefore:

$\text{final height}=\sin(A+B)$

But we have just calculated the final height as:

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

Therefore:

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

And we have derived the formula.

10. Why are the two terms added?

This is the key conceptual question.

At first, it can seem as though we should simply start with:

$\sin A$

and then add:

$\cos A\sin B$

because that is the new vertical contribution from the horizontal component.

But that would ignore something important.

While the horizontal component is gaining vertical height, the original vertical component is losing some of its vertical height.

So the actual calculation is:

starting height + gain − loss

In symbols:

$\sin A+\cos A\sin B-\sin A(1-\cos B)$

When simplified, this becomes:

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

So the two terms in the final formula are not simply two arbitrary contributions that we decided to add.

They appear because the original two components undergo two simultaneous changes.

11. A useful way to visualize the whole process

Think of the vertical height as a quantity that starts at:

$\sin A$

Then the rotation produces two effects.

Effect 1 — new vertical height

The horizontal component swings upward.

It contributes:

$\cos A\sin B$

Effect 2 — lost vertical height

The original vertical component tilts away from the vertical.

It loses:

$\sin A(1-\cos B)$

Therefore:

$\text{new height}=\text{old height}+\text{gain}-\text{loss}$

or:

$\sin(A+B)=\sin A+\cos A\sin B-\sin A(1-\cos B)$

Simplifying:

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

12. What the auxiliary circles are really doing

The auxiliary circles in the image are a technique for measuring the two effects.

We don’t need to physically rotate the original horizontal and vertical line segments.

Instead:

For the horizontal component

Take its length:

$\cos A$

Use it as the radius of an auxiliary circle.

Rotate that radius through B.

Its vertical contribution is:

$\cos A\sin B$

For the vertical component

Take its length:

$\sin A$

Use it as the radius of another auxiliary circle.

Rotate that radius through B.

Its remaining vertical contribution is:

$\sin A\cos B$

This makes the geometry much easier to see.

13. The deeper lesson

The addition formula is therefore telling us something about components.

A unit radius at angle A consists of:

cos A horizontally

and:

sin A vertically

When the direction changes by B, both components participate in determining the new vertical coordinate.

The horizontal component contributes upward.

The vertical component remains partly vertical.

Together, they determine the final height.

That final height is exactly:

$\sin(A+B)$

Hence:

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

The technique in one sentence

Start with the original vertical height, identify how the original horizontal component adds vertical height and how the original vertical component loses vertical height, and combine those changes to obtain the vertical coordinate at A+B.

This way of thinking is often more intuitive than memorizing the addition formula, because the two terms emerge from the geometry rather than appearing as unexplained algebra.

Understanding sin(A + B) by Tracking the Change in Vertical Height

1. Start at angle A

2. Now rotate the radius through B

3. There are two effects during the rotation

4. Effect of the original horizontal component

5. Effect of the original vertical component