The formula
cos(x + y) = cos x cos y − sin x sin y
is often introduced as something that students simply need to memorize.
But there is a much more intuitive way to understand it.
The formula can be derived by thinking about a vector, breaking that vector into its horizontal and vertical components, and then asking a simple question:
What does each original component contribute to the final horizontal length when the vector is rotated by an additional angle y?
The important insight is that we can examine the two components independently and then combine their contributions.
1. Start with a unit vector at angle x
Consider a unit vector pointing at an angle x from the positive horizontal axis.
Because its length is 1, its coordinates are:
(cos x, sin x)
Therefore, the vector has two components:
Horizontal component = cos x
Vertical component = sin x
We can visualize this as:
P
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/|
/ |
/ | sin x
/ |
/ |
/ |
/ x |
/ |
O────────●
cos x
The vector is therefore composed of:
- a horizontal component of length cos x
- a vertical component of length sin x
These are simply the two components of the same vector.
2. Now add an angle y
Suppose we rotate the entire vector counterclockwise through an additional angle y.
The original angle was:
x
The additional rotation is:
y
Therefore, the final angle is:
x + y
The final unit vector therefore has coordinates:
(cos(x + y), sin(x + y))
Our immediate goal is to find its horizontal component.
That horizontal component is:
cos(x + y)
Instead of trying to find it all at once, we will calculate the contribution from each of the original components separately.
3. First component: the original cos x
Let’s temporarily ignore the vertical component.
The original horizontal component has length:
cos x
Initially, it points directly to the right.
Now rotate this component through angle y.
It becomes a slanted vector.
B
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/
/ length = cos x
/
/ y
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We want to know:
How much of this rotated vector is still pointing horizontally?
The answer comes from a right triangle.
The vector itself is the hypotenuse, with length:
cos x
The horizontal projection is the side adjacent to angle y.
Using the definition of cosine:
cos y = adjacent side / hypotenuse
Therefore:
cos y = horizontal contribution / cos x
Multiplying by cos x:
horizontal contribution = cos x cos y
So the original horizontal component contributes:
+cos x cos y
to the final horizontal direction.
The contribution is positive because it points to the right.
4. Second component: the original sin x
Now let’s forget about the horizontal component and look only at the original vertical component.
Its length is:
sin x
Initially, it points straight upward.
↑
│
│
│ sin x
│
│
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At this stage, its horizontal contribution is zero.
Why?
Because the vector is vertical.
It makes an angle of 90° with the horizontal axis.
So initially:
horizontal contribution = 0
5. Now rotate the vertical component by y
We now rotate this vertical component counterclockwise through angle y.
Because it starts by pointing straight upward, counterclockwise rotation makes it tilt toward the left.
↖
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/
/
/
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This is an important observation.
The vertical component has now developed a horizontal component.
But that horizontal component points to the left.
We need to determine its magnitude.
6. Finding the horizontal contribution of sin x
The original vertical component has length:
sin x
After rotating it through y, it forms a right triangle.
The horizontal component is opposite the angle y.
Therefore:
sin y = horizontal component / sin x
Multiplying by sin x:
horizontal component = sin x sin y
So the magnitude of the horizontal component is:
sin x sin y
But remember that it points to the left.
Right is positive and left is negative.
Therefore its signed contribution is:
−sin x sin y
7. The two components are treated independently
We can now see the central idea.
We started with two original components:
cos x
and:
sin x
We examine what each one does independently when the rotation by y is applied.
Original horizontal component
cos x
becomes a horizontal contribution of:
+cos x cos y
Original vertical component
sin x
develops a horizontal contribution of:
−sin x sin y
We didn’t need to know the size of the other component while calculating either contribution.
This is the key insight.
8. Now combine the two contributions
The final horizontal component is simply the sum of the two horizontal contributions.
Therefore:
final horizontal component
= cos x cos y + (−sin x sin y)
or:
final horizontal component
= cos x cos y − sin x sin y
But after rotating the original vector by y, the final angle is:
x + y
And the horizontal component of a unit vector at angle x + y is:
cos(x + y)
Therefore:
cos(x + y) = cos x cos y − sin x sin y
This is the cosine addition formula.
9. The entire idea in one table
The derivation becomes especially clear if we put the two contributions side by side:
| Original component | What happens after rotation by y? | New horizontal contribution |
|---|---|---|
| cos x | Projects horizontally using cos y | +cos x cos y |
| sin x | Develops a leftward horizontal component using sin y | −sin x sin y |
Now add them:
+cos x cos y
−sin x sin y
Therefore:
cos(x + y) = cos x cos y − sin x sin y
10. Why can we treat the two components independently?
This is an important mathematical idea.
The original vector can be written as:
original vector = horizontal component + vertical component
That is:
original vector = cos x + sin x
where the two quantities are understood as perpendicular vector components.
When we rotate the vector, we can rotate those components separately:
rotated vector = rotated horizontal component + rotated vertical component
This works because vectors obey vector addition.
In other words, the effect of the rotation on the whole vector is exactly the combined effect of the rotation on its individual components.
So we are not making an approximation when we calculate the two contributions separately. We are using the fundamental properties of vectors.
11. The “not aware” intuition
There is a particularly useful way to think about this.
The original cos x component doesn’t need to know how large the sin x component is.
We can ask independently:
If I take a horizontal vector of length cos x and rotate it by y, how much horizontal length does it contribute?
Answer:
cos x cos y
Likewise, the original sin x component doesn’t need to know how large the cos x component is.
We ask:
If I take a vertical vector of length sin x and rotate it by y, how much horizontal length does it contribute?
Answer:
−sin x sin y
Only after calculating these two independent contributions do we combine them.
This is why the formula naturally takes the form:
cos x cos y − sin x sin y
12. Why does the minus sign appear?
The minus sign is not an arbitrary part of the formula.
It comes entirely from direction.
The first contribution points right:
+cos x cos y
The second contribution points left:
−sin x sin y
Therefore:
rightward contribution − leftward contribution
gives:
cos x cos y − sin x sin y
So the minus sign has a clear geometric meaning.
13. A simple numerical example
Suppose:
x = 30°
and:
y = 60°
Then:
x + y = 90°
We know:
cos 30° ≈ 0.866
sin 30° = 0.5
cos 60° = 0.5
sin 60° ≈ 0.866
The first component contributes:
cos x cos y
= 0.866 × 0.5
= 0.433
The second component contributes:
−sin x sin y
= −0.5 × 0.866
= −0.433
Combine them:
0.433 − 0.433 = 0
And indeed:
cos 90° = 0
Geometrically, this makes perfect sense.
After rotating the vector to 90°, it points completely upward, so there is no horizontal component.
The rightward and leftward contributions exactly cancel.
14. The same idea explains the sine addition formula
Once we understand the horizontal component, the sine formula becomes much easier.
Instead of asking:
What is the final horizontal component?
we ask:
What is the final vertical component?
The original horizontal component cos x produces an upward contribution:
+cos x sin y
The original vertical component sin x produces an upward contribution:
+sin x cos y
Both point upward, so they are added:
cos x sin y + sin x cos y
The final vertical component is:
sin(x + y)
Therefore:
sin(x + y) = sin x cos y + cos x sin y
So the two fundamental addition formulas arise from exactly the same principle.
15. The bigger picture
The entire derivation can be thought of as a three-step process:
Step 1 — Decompose
Start with the vector at angle x:
Horizontal = cos x
Vertical = sin x
Step 2 — Rotate each component
The additional rotation is y.
The original horizontal component contributes:
+cos x cos y
The original vertical component contributes:
−sin x sin y
Step 3 — Combine
Add the horizontal contributions:
cos x cos y − sin x sin y
This must equal the final horizontal component:
cos(x + y)
Therefore:
cos(x + y) = cos x cos y − sin x sin y
16. Why this is more powerful than memorization
Once we understand this derivation, the formula no longer looks like four unrelated terms.
Each term has a job:
cos x → size of the original horizontal component
cos y → horizontal fraction of that component after rotation
sin x → size of the original vertical component
sin y → horizontal fraction produced by rotating that vertical component
The minus sign → the second contribution points left.
So:
cos x cos y − sin x sin y
is really saying:
Final horizontal length = contribution from the original horizontal part − contribution from the original vertical part.
And because the final direction is x + y, that final horizontal length is:
cos(x + y).
Thus:
cos(x + y) = cos x cos y − sin x sin y
This way of understanding the formula makes the later formulas for cos 2x, cos 3x, cos 4x, and higher multiple angles much easier to derive rather than memorize.
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