When we study trigonometry, we encounter two extremely important formulas:
sin(x + y) = sin x cos y + cos x sin y
and
cos(x + y) = cos x cos y − sin x sin y
These are called the angle-addition formulas.
They are much more important than formulas such as:
sin 2x
cos 2x
sin 3x
cos 3x
because those formulas can be derived from the angle-addition formulas.
But there is an even more important question:
Why do the angle-addition formulas have this particular structure?
To understand that, we can think about vectors, components, rotations, and right triangles.
1. Start with a vector at angle x
Imagine a vector of length 1 pointing at an angle x from the positive x-axis.
Because the vector has length 1, its coordinates are:
(cos x, sin x)
This means:
Horizontal component = cos x
Vertical component = sin x
We can therefore think of the vector as being made from two perpendicular components:
- a horizontal component of length cos x
- a vertical component of length sin x
The important thing is that these are not two different vectors originally. They are simply the horizontal and vertical parts of the same vector.
2. Now rotate the vector by another angle y
Suppose we rotate the entire vector counterclockwise through an additional angle y.
The original vector was pointing at:
x
After rotating it by y, it points at:
x + y
Therefore, the new vector has coordinates:
(cos(x + y), sin(x + y))
We want to understand how these new coordinates are produced.
The key insight is:
The original horizontal and vertical components both contribute to the new horizontal and vertical components after the rotation.
Let’s examine the horizontal component first.
3. What happens to the original horizontal 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.
Suppose, for the moment, that its length is some general value L.
After rotating it through y, we can draw a right triangle:
B
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/ |
/ |
/ |
/ |
/ |
/ y |
/ |
O────────A
horizontal component
The original vector OB is the hypotenuse.
Its length is:
L
The horizontal component OA is the side adjacent to angle y.
From the definition of cosine:
cos y = adjacent side / hypotenuse
Therefore:
cos y = horizontal component / L
Rearranging:
horizontal component = L cos y
This is the important geometric fact.
4. What does “cos y gives the fraction” really mean?
We often say:
“cos y tells us what fraction of the vector remains horizontal.”
This is simply another way of saying:
cos y = horizontal component / vector length
For example, suppose a vector has length 10.
If we rotate it through 60°:
cos 60° = 1/2
Therefore:
horizontal component = 10 × 1/2
So:
horizontal component = 5
In other words, half of the vector’s length contributes to the horizontal direction.
We are not saying that the vector itself becomes shorter.
The vector is still length 10.
We are saying that its projection onto the horizontal axis is 5.
This distinction is important.
5. Apply this to the original horizontal component
Our original horizontal component has length:
cos x
So take:
L = cos x
After rotating it by y, its horizontal contribution becomes:
cos x × cos y
Therefore:
horizontal contribution from the original horizontal component = cos x cos y
This gives the first part of the cosine addition formula.
6. Now consider the original vertical component
The original vertical component has length:
sin x
Initially, it points straight upward.
Now rotate this component counterclockwise through angle y.
Something interesting happens.
It tilts toward the left.
Therefore, after the rotation, it develops a horizontal component pointing in the negative x-direction.
The magnitude of that horizontal component is:
sin x sin y
Why is sine used here?
Because the original vector is vertical.
After rotating it by y, the horizontal component forms the side opposite the angle y.
Recall:
sin y = opposite side / hypotenuse
Therefore:
horizontal component = vector length × sin y
The vector’s length is:
sin x
So:
horizontal component = sin x sin y
But this component points to the left, not the right.
Therefore its signed horizontal contribution is:
−sin x sin y
7. Put the two horizontal contributions together
We now have two contributions to the final horizontal component.
From the original horizontal component:
+cos x cos y
From the original vertical component:
−sin x sin y
Therefore:
final horizontal component = cos x cos y − sin x sin y
But after the rotation, the entire vector is pointing at angle:
x + y
For a unit vector at angle x + y, the horizontal component is:
cos(x + y)
Therefore:
cos(x + y) = cos x cos y − sin x sin y
And this is the cosine addition formula.
8. Why the minus sign is now understandable
The minus sign is not something that has to be memorized independently.
It comes from the direction of the horizontal contribution.
The original horizontal component produces a contribution pointing:
right → positive
The original vertical component, after counterclockwise rotation, produces a contribution pointing:
left → negative
Therefore:
right contribution − left contribution
which gives:
cos x cos y − sin x sin y
So the minus sign has a geometric meaning.
9. Now derive the sine addition formula
We can use exactly the same reasoning for the vertical component.
The original horizontal component has length:
cos x
After rotating it through y, its vertical contribution is:
cos x sin y
This contribution points upward.
Now consider the original vertical component.
It has length:
sin x
After rotating it through y, its vertical contribution becomes:
sin x cos y
This also points upward.
Therefore, the two vertical contributions are added:
cos x sin y + sin x cos y
But the final vertical component is:
sin(x + y)
Therefore:
sin(x + y) = cos x sin y + sin x cos y
Rearranging the terms:
sin(x + y) = sin x cos y + cos x sin y
This is the sine addition formula.
10. We can summarize the entire geometry
The original vector has:
Horizontal component = cos x
Vertical component = sin x
After rotating through y:
Horizontal direction
The original horizontal part contributes:
cos x cos y
The original vertical part contributes:
−sin x sin y
Therefore:
cos(x + y) = cos x cos y − sin x sin y
Vertical direction
The original horizontal part contributes:
cos x sin y
The original vertical part contributes:
sin x cos y
Therefore:
sin(x + y) = sin x cos y + cos x sin y
This gives us both fundamental formulas.
11. A useful component table
The whole derivation can be organized into a simple table:
| Original component | New horizontal contribution | New vertical contribution |
|---|---|---|
| cos x | cos x cos y | cos x sin y |
| sin x | −sin x sin y | sin x cos y |
Now add the contributions in each direction.
Horizontal:
cos x cos y − sin x sin y
Therefore:
cos(x + y) = cos x cos y − sin x sin y
Vertical:
cos x sin y + sin x cos y
Therefore:
sin(x + y) = sin x cos y + cos x sin y
12. Why these formulas are fundamental
These two identities are more important than many of the formulas we normally memorize.
The reason is that we can use them to generate other formulas.
For example, to find sin 2x, notice:
2x = x + x
Therefore:
sin 2x = sin(x + x)
Using the addition formula:
sin 2x = sin x cos x + cos x sin x
Therefore:
sin 2x = 2 sin x cos x
We have derived the double-angle formula.
13. Deriving cos 2x
Similarly:
2x = x + x
Therefore:
cos 2x = cos(x + x)
Using the addition formula:
cos 2x = cos x cos x − sin x sin x
So:
cos 2x = cos²x − sin²x
Again, the double-angle formula is simply a special case of the addition formula.
14. Deriving sin 3x
We can write:
3x = 2x + x
Therefore:
sin 3x = sin(2x + x)
Using the addition formula:
sin 3x = sin 2x cos x + cos 2x sin x
We already know:
sin 2x = 2 sin x cos x
and:
cos 2x = cos²x − sin²x
Substituting and simplifying eventually gives:
sin 3x = 3 sin x − 4 sin³x
So even the triple-angle formula comes from the same fundamental identity.
15. Deriving cos 3x
Similarly:
3x = 2x + x
so:
cos 3x = cos(2x + x)
Using the cosine addition formula and simplifying gives:
cos 3x = 4 cos³x − 3 cos x
Again, no new principle was required.
16. What about sin 4x and cos 4x?
Exactly the same idea works.
We can write:
4x = 2x + 2x
Therefore:
sin 4x = sin(2x + 2x)
and:
cos 4x = cos(2x + 2x)
Using the addition formulas gives:
sin 4x = 4 sin x cos x(cos²x − sin²x)
and:
cos 4x = 8 cos⁴x − 8 cos²x + 1
So yes, we can continue this process for higher multiples.
17. The general pattern
Suppose we want to find sin nx or cos nx.
We can write:
nx = (n − 1)x + x
Then:
sin(nx) = sin((n − 1)x + x)
and:
cos(nx) = cos((n − 1)x + x)
The addition formulas allow us to build the next formula from the previous ones.
Thus:
sin x, cos x
lead to:
sin 2x, cos 2x
which lead to:
sin 3x, cos 3x
which lead to:
sin 4x, cos 4x
and so on.
18. Connection with our parametric-curve problem
This idea is exactly what helped us simplify the parametric curve:
x = 3 cos t + cos 3t
y = 3 sin t − sin 3t
We used:
cos 3t = 4 cos³t − 3 cos t
and:
sin 3t = 3 sin t − 4 sin³t
Therefore:
x = 3 cos t + 4 cos³t − 3 cos t
so:
x = 4 cos³t
Similarly:
y = 3 sin t − (3 sin t − 4 sin³t)
so:
y = 4 sin³t
Thus the apparently complicated parametric equations are actually describing the much simpler curve:
x = 4 cos³t
y = 4 sin³t
This is an astroid.
And the triple-angle formulas that made this simplification possible ultimately came from the fundamental angle-addition formulas.
19. The deeper idea
There is a very useful conceptual chain here:
A vector has horizontal and vertical components
↓
Sine and cosine describe those components
↓
Rotating the vector changes both components
↓
Each old component contributes to both new components
↓
Using right-triangle relationships gives the new contributions
↓
Adding those contributions produces the angle-addition formulas
Therefore:
cos(x + y) = cos x cos y − sin x sin y
and:
sin(x + y) = sin x cos y + cos x sin y
These are not arbitrary algebraic formulas.
They are really statements about how the horizontal and vertical components of a vector change when we combine rotations.
20. The main lesson
Instead of memorizing a long list of trigonometric formulas, it is much more powerful to understand these two:
sin(x + y) = sin x cos y + cos x sin y
cos(x + y) = cos x cos y − sin x sin y
Once these are understood, many other formulas can be reconstructed whenever we need them.
The most important geometric idea to remember is:
When a vector is rotated, its original horizontal and vertical components both contribute to the new horizontal and vertical components.
For the horizontal direction:
cos x cos y − sin x sin y
For the vertical direction:
cos x sin y + sin x cos y
And because the final angle is x + y, these become:
cos(x + y)
and
sin(x + y).
That is the fundamental idea from which the double-angle, triple-angle, quadruple-angle, and higher multiple-angle formulas can all be derived.
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