A small rotation of just 1° can help us understand one of the most confusing ideas in trigonometry: how a positive sine value can be associated with a negative horizontal coordinate.
The apparent contradiction disappears once we distinguish between magnitude and direction.
Q1. Where do we start?
Start with a unit vector pointing straight upward.
Its angle from the positive x-axis is:
90°
Its coordinates are:
(0, 1)
So we can draw it as:
↑
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O
There is:
- no horizontal component
- a positive vertical component of 1
Q2. What happens if we rotate it anticlockwise by only 1°?
Now rotate the vector anticlockwise through 1°.
Its new angle is:
90° + 1° = 91°
The vector now points slightly to the left of the vertical:
↑ 90°
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O
↖ 91°
It is still pointing mostly upward, but it has moved slightly toward the left.
Q3. What are the coordinates of the new vector?
Because the vector is now at 91°, its coordinates are:
(cos 91°, sin 91°)
We know approximately:
cos 91° = −0.01745
and
sin 91° = +0.99985
Therefore:
91° vector = (−0.01745, 0.99985)
This tells us:
- horizontal coordinate = −0.01745
- vertical coordinate = +0.99985
So the vector points:
slightly left and almost completely upward.
Q4. But isn’t sin 1° positive?
Yes!
This is the important observation.
sin 1° ≈ +0.01745
There is absolutely nothing negative about sin 1°.
So why does the horizontal coordinate become:
−0.01745?
Because:
cos 91° = −sin 1°
Therefore:
cos 91° = −0.01745
The sin 1° value tells us the size of the small horizontal deviation.
The minus sign tells us that this deviation is to the left.
Q5. Can we see this geometrically?
Yes.
Start with the vertical direction:
↑
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O
Rotate it anticlockwise by 1°:
↑
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| \
| ↖
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O
The vector has moved slightly to the left.
The small horizontal distance has magnitude:
sin 1°
But it lies to the left of the vertical axis.
Therefore its signed x-coordinate is:
−sin 1°
At the same time, the vertical distance is:
cos 1°
and it remains upward, so it is positive.
Thus the rotated vector has components:
(−sin 1°, cos 1°)
Q6. Why is the horizontal distance sin 1°?
Look at the tiny right triangle formed by the rotated vector.
The rotated vector has length 1.
The small angle between the original vertical direction and the rotated vector is:
1°
The horizontal side is opposite that 1° angle.
Therefore:
horizontal magnitude = sin 1°
The vertical side is adjacent to the 1° angle.
Therefore:
vertical magnitude = cos 1°
So the lengths are:
horizontal length = sin 1°
vertical length = cos 1°
But the horizontal length lies to the left, so its coordinate is:
−sin 1°
Q7. Is sin 1° negative?
No.
This distinction is essential:
sin 1° = +0.01745
The horizontal coordinate is:
−sin 1° = −0.01745
These are not contradictory statements.
The first is a positive magnitude.
The second is a signed coordinate.
Q8. What is the difference between a length and a coordinate?
Suppose you walk 1 metre.
The distance travelled is:
1 metre
It is positive.
But if you define right as positive and you walk 1 metre to the left, your signed displacement is:
−1 metre
The physical distance did not become negative.
The minus sign tells you the direction.
The same principle applies here.
Magnitude
sin 1° = 0.01745
Direction
left
Signed horizontal coordinate
−sin 1° = −0.01745
Q9. How does this lead to the general rotation formula?
Now replace 1° with an arbitrary angle B.
Start with the upward unit direction:
(0,1)
Rotate it anticlockwise through B.
Its new direction is:
90° + B
Therefore its coordinates are:
(cos(90°+B), sin(90°+B))
Using the standard angle relationships:
cos(90°+B) = −sin B
and
sin(90°+B) = cos B
So:
(0,1) → (−sin B, cos B)
This is not saying that sin B is negative.
It says:
The horizontal displacement has magnitude
sin B, but because it is to the left, its x-coordinate is−sin B.
Q10. What happens if B gets larger?
The 1° example is particularly useful because the geometry is easy to see.
But the same principle continues as B increases.
B = 1°
The vector points slightly left of vertical:
(−sin 1°, cos 1°)
B = 30°
It points substantially left but still upward:
(−sin 30°, cos 30°)
B = 90°
It points completely left:
(−1, 0)
At this point:
sin 90° = 1
and:
cos 90° = 0
So:
(−sin 90°, cos 90°) = (−1, 0)
The vector has rotated from upward to left.
Q11. What happens after 90°?
Now something interesting happens.
Suppose:
B = 91°
Then:
−sin 91° < 0
so the horizontal component remains leftward.
But:
cos 91° < 0
so the vertical component has now become downward.
Therefore:
(−sin 91°, cos 91°)
points:
left + down
which is exactly what we expect.
The trigonometric functions automatically provide the changing signs.
Q12. So what is the real source of the minus sign?
The minus sign does not mean:
“sin B is negative.”
Instead:
sin B gives the positive magnitude of the horizontal deviation from the vertical, while the minus sign tells us that this deviation is toward the negative x-direction.
For the small 1° rotation:
sin 1° = positive magnitude
−sin 1° = leftward signed coordinate
That is the cleanest interpretation.
Q13. Why is the 1° example so useful?
Because it separates two ideas that can easily become confused.
At 91°:
sin 91° is positive
because the point is still above the x-axis.
But:
cos 91° is negative
because the point is slightly to the left of the y-axis.
The coordinates are therefore:
(negative, positive)
This tells us the point is in the second quadrant.
So:
The sign of sine tells us about the vertical coordinate. The sign of cosine tells us about the horizontal coordinate.
That is why:
sin 91° > 0
while:
cos 91° < 0
Q14. What does this teach us about trigonometry more generally?
It shows why we should be careful when moving from right-triangle geometry to arbitrary angles.
In a right triangle, side lengths are normally treated as positive magnitudes.
On the coordinate plane, components are signed quantities.
Sine and cosine connect these two ideas.
For an arbitrary angle:
- sine describes the vertical coordinate of the unit-circle point;
- cosine describes the horizontal coordinate of the unit-circle point.
Their signs therefore tell us the quadrant.
The Key Insight
sin 1° is positive. It does not point left.
A number does not have a direction by itself.
When we rotate the upward direction anticlockwise by 1°, the resulting vector moves slightly to the left.
The size of that leftward movement is:
sin 1°
Because the movement is leftward, its signed horizontal coordinate is:
−sin 1°
Thus:
cos 91° = −sin 1°
while:
sin 91° = cos 1°
and the vector at 91° is:
(−sin 1°, cos 1°)
This distinction between positive magnitude and signed coordinate is one of the most useful ideas for understanding why trigonometric formulas continue to work when angles move beyond the first quadrant.
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