The trigonometric addition formulas can seem like algebraic rules that we simply have to memorize.
But geometrically, something much more satisfying is happening.
We begin with a radius on a unit circle, break its coordinates into horizontal and vertical components, account for an additional rotation, and finally reconstruct the coordinates of the new point on the same unit circle.
The formulas are simply the numerical expression of that geometric process.
1. Start with the unit circle
Place a circle of radius 1 at the origin of a Cartesian coordinate system.
y
↑
.-------.
.-' '-.
.' '.
/
| O----------→ x
/
'. .'
'-. .-'
'-------'
The radius of this circle is exactly:
1
This is why it is called the unit circle.
2. Choose a radius at angle A
Now draw a radius from the origin making an angle A with the positive x-axis.
y
↑
|
| ●
| /
| /
| /
-----------------O---/------------→ x
A
Because the radius has length 1, the endpoint has coordinates:
(cos A, sin A)
Therefore:
cos A = x-component
sin A = y-component
So the radius at A can be described through its two Cartesian components:
x = cos A
y = sin A
3. Now rotate the radius by B
Suppose we rotate the radius anticlockwise through another angle B.
The final angle is:
A + B
We therefore arrive at a new point on the same unit circle.
y
↑
|
| ● A+B
| /
| /
| ● A
| /
-----------------O----------------→ x
The important question is:
What are the x and y coordinates of this new point?
By the definition of sine and cosine on the unit circle, those coordinates must be:
x = cos(A+B)
y = sin(A+B)
But we want to understand how we can obtain them from the original components at A.
4. The original radius already has two components
The radius at A has:
horizontal component = cos A
vertical component = sin A
These two components together describe the original unit radius.
Now the entire radius is rotated through B.
Instead of treating the radius as one mysterious object, we can track what happens to each of its two components.
5. What happens to the original horizontal component?
The original horizontal component has length:
cos A
After rotating it through B, it is no longer purely horizontal.
It produces two Cartesian contributions:
rotated component
/
/
/
-----------------→
Its contributions are:
horizontal contribution = cos A cos B
vertical contribution = cos A sin B
So the original horizontal component contributes to both final coordinates.
6. What happens to the original vertical component?
The original vertical component has length:
sin A
When it is rotated through B, it also becomes tilted.
Its contributions are:
vertical contribution = sin A cos B
and
horizontal contribution = −sin A sin B
The minus sign occurs because this horizontal contribution points toward the left.
Thus the original vertical component also contributes to both final coordinates.
7. Now reconstruct the final x-coordinate
The final x-coordinate receives two contributions.
From the original horizontal component:
cos A cos B
From the original vertical component:
−sin A sin B
Adding them gives:
cos A cos B − sin A sin B
But what is this final x-coordinate?
We already know that the final radius has angle A+B.
On the unit circle, the x-coordinate at angle A+B is:
cos(A+B)
Therefore:
cos(A+B) = cos A cos B − sin A sin B
8. Now reconstruct the final y-coordinate
The final y-coordinate also receives two contributions.
From the original horizontal component:
cos A sin B
From the original vertical component:
sin A cos B
Adding them gives:
cos A sin B + sin A cos B
But the final y-coordinate at angle A+B is:
sin(A+B)
Therefore:
sin(A+B) = sin A cos B + cos A sin B
9. The final result is still on the unit circle
This is perhaps the most beautiful part of the derivation.
We did not create a new circle.
We did not move to a different coordinate system.
We started with:
one Cartesian coordinate system and one unit circle.
We began with the radius at A.
We accounted for an additional rotation B.
And we ended at the point corresponding to A+B on that same unit circle.
The final coordinates are:
x = cos(A+B)
y = sin(A+B)
And our component calculation tells us exactly what those coordinates are:
x = cos A cos B − sin A sin B
y = sin A cos B + cos A sin B
Therefore:
cos(A+B) = cos A cos B − sin A sin B
sin(A+B) = sin A cos B + cos A sin B
10. What the formulas are really saying
The formulas can now be understood as statements about coordinates.
Cosine addition formula
The x-coordinate of the radius at A+B is obtained by adding all the horizontal contributions created when the original components at A are rotated through B.
That gives:
cos(A+B) = cos A cos B − sin A sin B
Sine addition formula
The y-coordinate of the radius at A+B is obtained by adding all the vertical contributions created when the original components at A are rotated through B.
That gives:
sin(A+B) = sin A cos B + cos A sin B
11. The entire derivation in one chain
The complete geometric process can be summarized as:
UNIT CIRCLE
↓
Radius at A
↓
Coordinates (cos A, sin A)
↓
Rotate through B
↓
Original components
are now tilted
↓
Calculate their x and y
contributions
↓
Add them
↓
Final coordinates
(cos(A+B), sin(A+B))
↓
Same unit circle
So the addition formulas are not something that suddenly appears at the end.
They emerge naturally from the coordinate reconstruction.
12. The deeper geometric insight
The most important idea is this:
A point on the unit circle is completely described by the x and y components of its radius.
At angle A:
(cos A, sin A)
After rotating through B:
(cos(A+B), sin(A+B))
The addition formulas explain how the original x and y components transform when the radius is rotated by an additional angle B.
So we can think of the entire derivation as a journey:
A → components → rotation by B → new components → A+B
And because the radius always remains a unit radius, the final result naturally brings us back to the unit circle.
Key Insight
The trigonometric addition formulas are fundamentally coordinate formulas for a rotated radius on the unit circle.
We start with:
(cos A, sin A)
Rotate through B.
Then reconstruct the new coordinates:
(cos A cos B − sin A sin B,
sin A cos B + cos A sin B)
But those coordinates belong to the point at angle A+B on the unit circle.
Therefore:
(cos(A+B), sin(A+B))
must equal:
(cos A cos B − sin A sin B,
sin A cos B + cos A sin B)
And from the equality of the corresponding coordinates, the two addition formulas follow.
We start on the unit circle, transform the coordinates, and finish on the same unit circle.
That is the geometric story behind the formulas.
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