We have already seen that the angle-addition formulas allow us to derive formulas for angles such as 2t, 3t, 4t, and beyond.
But what happens when we subtract an angle?
For example:
- sin(x − y)
- cos(x − y)
At first, these may appear to require two completely new formulas. In fact, they come directly from the same angle-addition formulas.
The key idea is:
Subtraction can be treated as addition of a negative angle.
1. The two fundamental angle-addition formulas
We begin with:
sin(A + B) = sin A cos B + cos A sin B
and
cos(A + B) = cos A cos B − sin A sin B
These are the fundamental formulas from which the subtraction formulas can be derived.
2. Turning subtraction into addition
Suppose we have:
x − y
We can rewrite this as:
x − y = x + (−y)
This is the crucial step.
Instead of trying to create a completely new formula for subtraction, we simply use the existing addition formula with −y.
3. Deriving sin(x − y)
Start with:
sin(x − y)
Rewrite the subtraction:
sin(x − y) = sin[x + (−y)]
Now apply the sine addition formula:
sin[x + (−y)] = sin x cos(−y) + cos x sin(−y)
We now need to understand what happens to sine and cosine when the angle is negative.
We know:
cos(−y) = cos y
and
sin(−y) = −sin y
Therefore:
sin(x − y) = sin x cos y + cos x(−sin y)
Hence:
sin(x − y) = sin x cos y − cos x sin y
So the subtraction formula is:
sin(x − y) = sin x cos y − cos x sin y
4. Why does cos(−y) = cos y?
This has a simple geometric explanation.
On the unit circle, cosine represents the horizontal coordinate.
Consider two angles, y and −y.
They are mirror images of each other across the x-axis.
y
/
/
•
/
---------O----------------→
\
\
•
\
-y
The two points have the same horizontal coordinate.
Therefore:
cos(−y) = cos y
However, their vertical coordinates have opposite signs.
Therefore:
sin(−y) = −sin y
This is why cosine is called an even function, while sine is called an odd function.
5. Deriving cos(x − y)
Now let’s do the same thing for cosine.
Start with:
cos(x − y)
Rewrite subtraction as addition:
cos(x − y) = cos[x + (−y)]
Apply the cosine addition formula:
cos[x + (−y)] = cos x cos(−y) − sin x sin(−y)
Use:
cos(−y) = cos y
and
sin(−y) = −sin y
Therefore:
cos(x − y) = cos x cos y − sin x(−sin y)
The two negative signs cancel:
cos(x − y) = cos x cos y + sin x sin y
Thus:
cos(x − y) = cos x cos y + sin x sin y
6. Compare addition and subtraction
We can now put all four formulas together.
Sine
sin(x + y) = sin x cos y + cos x sin y
sin(x − y) = sin x cos y − cos x sin y
Cosine
cos(x + y) = cos x cos y − sin x sin y
cos(x − y) = cos x cos y + sin x sin y
There is a very interesting pattern here.
When we change +y to −y, the sign between the two terms changes.
7. Why does the sign change?
There is also a geometric way to understand this.
Think of an angle x as the direction of a vector.
If we add y, we rotate the vector counterclockwise:
x → x + y
If we subtract y, we rotate the vector clockwise:
x → x − y
So the two operations are opposite rotations.
x + y
↗
/
/
/
/ x
--------------O----------------
\
\
↘
x − y
The direction of the rotation changes which component points to the left or right.
That change in direction is reflected algebraically by the change of signs in the formulas.
8. A useful way to remember the formulas
Instead of memorizing four unrelated formulas, remember the two addition formulas:
sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B − sin A sin B
Then remember:
x − y = x + (−y)
along with:
cos(−y) = cos y
sin(−y) = −sin y
The subtraction formulas then follow automatically.
9. The four fundamental formulas
We can summarize everything as follows:
sin(x + y) = sin x cos y + cos x sin y
sin(x − y) = sin x cos y − cos x sin y
cos(x + y) = cos x cos y − sin x sin y
cos(x − y) = cos x cos y + sin x sin y
These four formulas form one of the most important foundations of trigonometry.
10. From these formulas, many others follow
Once we understand the angle-addition and subtraction formulas, we can derive many other identities.
For example, setting y = x gives:
sin(x + x) = sin x cos x + cos x sin x
Therefore:
sin 2x = 2 sin x cos x
Similarly:
cos(x + x) = cos x cos x − sin x sin x
giving:
cos 2x = cos²x − sin²x
And by repeatedly adding x, we can obtain:
sin 3x
cos 3x
sin 4x
cos 4x
and so on.
So the formulas for multiple angles and the formulas for angle subtraction are not separate topics.
They all come from the same central idea:
Sine and cosine tell us the coordinates of a rotated vector, and the angle-addition formulas describe how those coordinates change when we rotate by another angle.
Once that idea is understood, the many trigonometric formulas become different consequences of one underlying principle rather than a long list of formulas to memorize.
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