When we first encounter the identities

cos(−t) = cos t

and

sin(−t) = −sin t

they can seem surprising.

It is natural to think:

“If t changes to −t, shouldn’t both sine and cosine change their signs?”

The answer becomes clear when we understand what sine and cosine represent geometrically.

The key idea is:

Cosine represents the horizontal coordinate of a point on the unit circle, while sine represents the vertical coordinate.

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1. Start with the unit circle

Consider a circle with radius 1 centered at the origin.

If a point P on the circle is located at angle t, its coordinates are:

(cos t, sin t)

This gives us an important interpretation:

• cos t = horizontal coordinate
• sin t = vertical coordinate

For an angle in the first quadrant, the situation looks like this:

                 y
                 ↑
                 |
                 |       P
                 |      •
                 |     /
                 |    /
                 |   / t
-----------------O----------------→ x

The point P has:

• a positive horizontal coordinate, cos t
• a positive vertical coordinate, sin t

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2. What does −t mean?

A positive angle t means rotating counterclockwise from the positive x-axis.

A negative angle −t means rotating clockwise from the positive x-axis.

So instead of going upward from the x-axis, we go downward:

                 y
                 ↑
                 |
                 |       P
                 |      •
                 |     /
                 |    / t
-----------------O----------------→ x
                 |    \
                 |     \
                 |      • Q
                 |       −t

The points P and Q are mirror images of each other across the x-axis.

This is the crucial geometric observation.

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3. What happens to the horizontal coordinate?

Suppose the point at angle t has coordinates:

(cos t, sin t)

After reflecting it across the x-axis, the point at angle −t has coordinates:

(cos t, −sin t)

Notice what happened.

The horizontal coordinate did not change.

For example:

       P = (0.8, 0.6)
              •

              |
--------------+------------→
              |
              •
       Q = (0.8, −0.6)

Both points are at the same horizontal position.

Therefore:

cos(−t) = cos t

Cosine does not change sign because the horizontal coordinate has not changed.

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4. What happens to the vertical coordinate?

The vertical coordinate is different.

The point P has vertical coordinate:

sin t

But after reflection across the x-axis, Q is below the x-axis.

Its vertical coordinate is therefore negative:

−sin t

So:

sin(−t) = −sin t

Thus, sine changes sign because the vertical coordinate changes sign.

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5. A numerical example

Let’s take:

t = 30°

We know:

cos 30° = √3/2

and

sin 30° = 1/2

Now consider −30°.

The point is reflected across the x-axis.

Therefore:

cos(−30°) = √3/2

but:

sin(−30°) = −1/2

So:

cos(−30°) = cos 30°

while:

sin(−30°) = −sin 30°

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6. Why don’t both coordinates change sign?

This is where the geometric picture is especially important.

Changing t to −t does not mean that we reverse the entire vector.

Instead, we change the direction of rotation.

Compare the two vectors:

                     +t
                    ↗
                   /
                  /
-----------------O----------------→ x
                  \
                   \
                    ↘
                     −t

The two vectors point in different directions, but they are mirror images across the x-axis.

Their:

• horizontal components are equal
• vertical components are opposite

Therefore:

horizontal component → unchanged

vertical component → sign reversed

Since:

cos t = horizontal component

and

sin t = vertical component

we obtain:

cos(−t) = cos t

sin(−t) = −sin t

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7. Another way to think about it

Imagine a point with coordinates:

(0.8, 0.6)

Reflect it across the x-axis.

The new point is:

(0.8, −0.6)

Only the y-coordinate changes sign.

Why?

Because reflection across the x-axis moves a point:

up → down

but does not move it:

left → right

Therefore:

Quantity	At t	At −t
Horizontal coordinate	0.8	0.8
Vertical coordinate	0.6	−0.6
Cosine	cos t	cos t
Sine	sin t	−sin t

This is exactly the behavior of cosine and sine.

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8. Why this matters for the subtraction formulas

This idea is particularly important when deriving the formulas for sin(x − y) and cos(x − y).

We write:

x − y = x + (−y)

Then the angle-addition formulas give:

sin(x − y) = sin x cos(−y) + cos x sin(−y)

Using:

cos(−y) = cos y

and

sin(−y) = −sin y

we obtain:

sin(x − y) = sin x cos y − cos x sin y

Similarly:

cos(x − y) = cos x cos(−y) − sin x sin(−y)

which becomes:

cos(x − y) = cos x cos y + sin x sin y

So the signs in the subtraction formulas are not arbitrary.

They ultimately come from the simple geometry of reflecting an angle across the x-axis.

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9. The deeper idea: even and odd functions

The identities

cos(−t) = cos t

and

sin(−t) = −sin t

also give us two important classifications.

Cosine is called an even function because:

f(−t) = f(t)

Sine is called an odd function because:

f(−t) = −f(t)

But these names are much easier to understand if we remember the geometry:

Cosine is unchanged under reflection across the x-axis, while sine changes sign.

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10. The simplest way to remember it

Don’t just memorize:

cos(−t) = cos t

sin(−t) = −sin t

Instead, remember:

Cosine is horizontal. Sine is vertical.

Changing t to −t reflects the point across the x-axis.

Reflection across the x-axis:

• leaves the horizontal coordinate unchanged
• reverses the vertical coordinate

Therefore:

cos(−t) = cos t

sin(−t) = −sin t

Once this geometric idea is clear, the signs in the angle-subtraction formulas become much more natural rather than something that has to be memorized.