The formulas
and
can initially look like formulas that simply have to be memorized.
But there is a geometric way to understand where these terms come from.
The key idea is:
The radius at angle A is not being replaced by two separate radii. It is one radius represented by two components: cos A horizontally and sin A vertically. When the radius is rotated through B, each component develops new horizontal and vertical contributions. We add contributions that lie in the same direction.
1. The original radius at angle A
Consider a unit circle with centre O.
Take a radius at angle A.
Its two components are:
horizontal component = cos A
vertical component = sin A
So we can represent the original radius as:
P
/|
/ |
/ | sin A
/ |
/ |
O─────H
cos A
The important point is:
and:
Here, the last expression should be understood as a vector decomposition into perpendicular components, not as ordinary scalar addition.
The horizontal component is:
and the vertical component is:
This is just another way of describing the same radius.
2. Now rotate the entire radius anticlockwise by B
The radius moves from angle A to:
We want to find the new horizontal and vertical components.
Instead of trying to calculate them directly, ask:
What happens to the original horizontal piece?
and:
What happens to the original vertical piece?
Then combine the results.
3. What happens to the original horizontal component?
The original horizontal component has length:
Imagine temporarily isolating that component.
It points horizontally:
O────────→
cos A
Now rotate it anticlockwise through B.
It becomes a sloping line:
↗
/ B
/
O
This rotated piece now has:
a horizontal component
and:
a vertical component
Because its original length was cos A:
Its horizontal contribution
The horizontal part is:
Its vertical contribution
The vertical part is:
So the original cos A has now produced two contributions:
cos A
↓ rotate by B
horizontal: cos A cos B
vertical: cos A sin B
4. Now what happens to the original vertical component?
Originally we had:
pointing vertically upward:
↑
│
│ sin A
│
O
Now rotate this component anticlockwise through B.
Because it starts vertically upward, it moves toward the left:
↖
/
/ B
O
Again, it now has two components.
Its original length is sin A.
Its vertical contribution
Because the angle B is measured from the vertical, the remaining vertical part is:
Its horizontal contribution
It points toward the left, so its horizontal contribution is:
Therefore:
sin A
↓ rotate by B
horizontal: −sin A sin B
vertical: sin A cos B
5. Now put the two rotated components together
This is the crucial step.
We have taken the same original radius and broken it into two components.
Original horizontal component
After rotation:
horizontal →
vertical →
Original vertical component
After rotation:
horizontal →
vertical →
Now we combine components in the same direction.
6. Find the final horizontal component
There are two horizontal contributions.
From the original horizontal component:
From the original vertical component:
Therefore:
But what is the final horizontal component?
The final radius is at angle:
and has length 1.
Therefore its horizontal component is:
So:
7. And the vertical component?
Exactly the same process.
There are two vertical contributions.
From the original horizontal component:
From the original vertical component:
Therefore:
The final radius is at angle A+B, so its vertical component is:
Therefore:
8. The complete picture
This is the part that makes the idea click:
ORIGINAL RADIUS
↗
/|
/ |
/ | sin A
/ |
/ |
O─────→
cos A
Rotate everything by B
But instead of rotating the whole radius as one mysterious object, imagine rotating its two components separately.
Original horizontal component
│
rotate by B
↓
horizontal: cos A cos B
vertical: cos A sin B
Original vertical component
│
rotate by B
↓
horizontal: −sin A sin B
vertical: sin A cos B
Then add the horizontal pieces together:
and add the vertical pieces together:
9. Why is this legitimate?
This is probably the deepest question behind the confusion.
You might ask:
“How can I take cos A and sin A separately and then add their changed components? Aren’t they really just parts of one radius?”
They are parts of one radius.
And that’s precisely why we can do this.
The original radius can be written as the sum of two perpendicular vectors:
horizontal component + vertical component
When a vector is rotated, each component rotates in exactly the same way.
And vector addition still works after the rotation.
So:
Rotate the whole vector
is equivalent to:
Rotate each component and then add the rotated components.
This is the fundamental geometric idea behind the formulas.
10. A very simple analogy
Imagine walking 3 metres east and 4 metres north.
Your overall displacement is:
3 east + 4 north
Now imagine turning your coordinate system by some angle.
The original 3-metre east component will now have:
some east-west contribution
and:
some north-south contribution
The original 4-metre north component will also have:
some east-west contribution
and:
some north-south contribution
To find your final east-west position, you add all east-west contributions.
To find your final north-south position, you add all north-south contributions.
That is exactly what is happening here.
11. The key insight
The radius at angle A is not being replaced by two separate radii.
It is one radius represented by two components:
cos A horizontally
and:
sin A vertically
When the radius is rotated through B, each component develops new horizontal and vertical contributions.
We then add contributions that point in the same direction.
Therefore:
Horizontal
So:
Vertical
So:
12. Final takeaway
The important idea is not simply to memorize the two formulas.
Instead, think:
Start with one radius at angle A. Break it into its horizontal and vertical components. Rotate the entire radius through B. Each original component now produces a new horizontal and vertical contribution. Add the contributions in each direction to obtain the final horizontal and vertical coordinates.
The final horizontal coordinate gives:
and the final vertical coordinate gives:
Therefore:
and:
This component-based viewpoint explains why the formulas contain two terms rather than making them look like formulas that have to be memorized.
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