When we first encounter the formula
cos(x + y) = cos x cos y − sin x sin y
it can look like a formula that simply has to be memorized.
But there is a much more intuitive way to understand it.
The key idea is to think about a unit vector, its horizontal and vertical components, and what happens to those components when the vector is rotated.
The remarkable result is that each of the two terms on the right-hand side comes from one of the original components of the vector.
1. Start with a unit vector at angle x
Consider a unit vector pointing at an angle x from the positive x-axis.
Because its length is 1, its coordinates are:
(cos x, sin x)
Therefore, the vector has:
Horizontal component = cos x
Vertical component = sin x
We can visualize this as a right triangle:
P
●
/|
/ |
/ | sin x
/ |
/ |
/ |
/ x |
/ |
O────────●
cos x
So the original vector can be thought of as being composed of two perpendicular pieces:
- a horizontal piece of length cos x
- a vertical piece of length sin x
Now comes the important part.
2. Rotate the entire vector by y
Suppose we rotate the vector counterclockwise through an additional angle y.
The original angle was:
x
After the rotation, the angle becomes:
x + y
Therefore, the final vector has coordinates:
(cos(x + y), sin(x + y))
We are going to concentrate first on its horizontal coordinate.
That horizontal coordinate is:
cos(x + y)
The question is:
Where does this final horizontal component come from?
The answer is that both original components contribute to it.
We will examine those two contributions separately.
3. First contribution: cos x cos y
Start with only the original horizontal component.
Its length is:
cos x
Initially, this component points directly to the right.
Now rotate this horizontal vector counterclockwise by y.
It becomes a slanted vector.
B
●
/
/
/ length = cos x
/
/ y
/
O────────────→
We want to know how much of this rotated vector points horizontally.
That is its horizontal projection.
4. Use the definition of cosine
The rotated vector has length:
cos x
The angle it makes with the horizontal direction is:
y
The horizontal component is the side adjacent to angle y.
Recall:
cos y = adjacent side / hypotenuse
Therefore:
cos y = horizontal component / cos x
Multiply both sides by cos x:
horizontal component = cos x cos y
So the original horizontal component contributes:
+cos x cos y
to the final horizontal direction.
The plus sign appears because this contribution points to the right, the positive x-direction.
5. Second contribution: −sin x sin y
Now forget the horizontal component for a moment.
Let’s look only at the original vertical component.
Its length is:
sin x
Initially, it points straight upward:
↑
│
│
│ sin x
│
│
O────────→ x
Now rotate this vertical vector counterclockwise through y.
Because it starts by pointing upward, counterclockwise rotation makes it tilt toward the left.
↖
/
/
/
/
/
O────────→ x
This is crucial.
The rotated vector now has a leftward horizontal component.
6. Find the magnitude of that horizontal component
The original vertical vector has length:
sin x
After rotating it by y, consider the right triangle formed by the rotated vector and its horizontal and vertical components.
The horizontal component is opposite the angle y.
Therefore, using the definition of sine:
sin y = opposite side / hypotenuse
Here:
opposite side = horizontal component
and:
hypotenuse = sin x
Therefore:
sin y = horizontal component / sin x
Multiply both sides by sin x:
horizontal component = sin x sin y
So the magnitude of the horizontal component is:
sin x sin y
But there is one more thing to consider: direction.
7. Why is it negative?
The horizontal component points to the left.
We normally define:
right = positive
and:
left = negative
Therefore, a leftward component of magnitude:
sin x sin y
has a signed value:
−sin x sin y
So the second contribution to the final horizontal component is:
−sin x sin y
This is exactly where the minus sign in the formula comes from.
8. Now combine the two contributions
We have considered the two original components independently.
First contribution
The original horizontal component produces:
+cos x cos y
Second contribution
The original vertical component produces:
−sin x sin y
Therefore, the total horizontal component is:
cos x cos y − sin x sin y
But what is the total horizontal component of the final vector?
The final vector makes an angle:
x + y
with the x-axis.
Since it is still a unit vector, its horizontal component is:
cos(x + y)
Therefore:
cos(x + y) = cos x cos y − sin x sin y
And we have derived the cosine addition formula.
9. The entire derivation in one picture
The idea can be summarized as follows:
Original vector
↗
/|
/ |
/ | sin x
/ |
/ |
/ |
O──────●
cos x
The vector contains two original components:
cos x horizontally
and:
sin x vertically.
After rotating both components by y:
Original horizontal component
length = cos x
↓
rotate by y
↓
horizontal contribution
+ cos x cos y
and:
Original vertical component
length = sin x
↓
rotate by y
↓
horizontal contribution
− sin x sin y
Now add them:
+cos x cos y
−sin x sin y
giving:
cos x cos y − sin x sin y
This is the final horizontal component.
Since the final angle is x + y, that horizontal component is cos(x + y).
Therefore:
cos(x + y) = cos x cos y − sin x sin y
10. Why the two products look different
It is worth pausing here because the structure of the formula now makes sense.
We have:
cos x cos y
and:
sin x sin y
Why these particular products?
Because each product has two parts.
For the first term:
cos x
is the length of the original horizontal component.
cos y
tells us what fraction of that component appears horizontally after rotation by y.
Therefore:
cos x × cos y
For the second term:
sin x
is the length of the original vertical component.
sin y
tells us the horizontal fraction produced when that vertical component is rotated by y.
Therefore:
sin x × sin y
The second contribution points left, so it receives a minus sign.
11. What about the sine formula?
The same reasoning also explains:
sin(x + y) = sin x cos y + cos x sin y
Instead of looking at the final horizontal component, look at the final vertical component.
The original vertical component contributes:
sin x cos y
and the original horizontal component contributes:
cos x sin y
This time, both contributions point upward, so they are added.
Therefore:
sin(x + y) = sin x cos y + cos x sin y
So the two fundamental formulas have a common geometric origin.
12. These formulas are the foundation for many others
Once we understand the angle-addition formulas, we don’t have to memorize every multiple-angle formula separately.
For example:
2x = x + x
so:
cos 2x = cos(x + x)
Using the addition formula:
cos 2x = cos²x − sin²x
Similarly:
3x = 2x + x
so:
cos 3x = cos(2x + x)
which eventually gives:
cos 3x = 4 cos³x − 3 cos x
And:
4x = 2x + 2x
allows us to derive cos 4x.
The same method works for sine.
So the fundamental angle-addition formulas are really the building blocks for the multiple-angle formulas.
13. The big picture
The derivation can be reduced to one simple idea:
A vector at angle x has two components: cos x horizontally and sin x vertically. When the vector is rotated by y, both original components contribute to the new horizontal component.
Those contributions are:
From the horizontal component:
+cos x cos y
From the vertical component:
−sin x sin y
Add them:
cos x cos y − sin x sin y
The final vector is at angle x + y, so its horizontal component is:
cos(x + y)
Therefore:
cos(x + y) = cos x cos y − sin x sin y
This is much more than a formula to memorize. It is a statement about how the components of a vector change when two rotations are combined.
And once this picture is understood, the double-angle, triple-angle, quadruple-angle, and higher multiple-angle formulas become natural consequences rather than a collection of unrelated formulas.
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