When we first encounter formulas such as

sin 3t = 3 sin t − 4 sin³t

and

cos 3t = 4 cos³t − 3 cos t

they can look like special formulas that we simply have to memorize.

But they are not special formulas at all.

They come directly from one fundamental idea:

The angle-addition formulas can generate every multiple-angle formula.

Once we understand this, formulas for 3t, 4t, 5t, 6t, and even higher multiples become part of one continuous process.

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1. Start with the angle-addition formulas

The two formulas we need are:

sin(A + B) = sin A cos B + cos A sin B

and

cos(A + B) = cos A cos B − sin A sin B

These formulas tell us what happens when we add two angles.

And a multiple angle is simply repeated addition of the same angle.

For example:

2t = t + t

3t = 2t + t

4t = 3t + t

5t = 4t + t

So we can repeatedly use the addition formulas.

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2. First derive the double-angle formulas

Before deriving 3t, we need the formulas for 2t.

Since

2t = t + t

we have:

sin 2t = sin(t + t)

Using the sine addition formula:

sin 2t = sin t cos t + cos t sin t

Therefore:

sin 2t = 2 sin t cos t

Similarly,

cos 2t = cos(t + t)

Using the cosine addition formula:

cos 2t = cos t cos t − sin t sin t

Therefore:

cos 2t = cos²t − sin²t

These double-angle formulas will now help us derive the triple-angle formulas.

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3. Deriving sin 3t

The key observation is:

3t = 2t + t

Therefore:

sin 3t = sin(2t + t)

Apply the angle-addition formula:

sin 3t = sin 2t cos t + cos 2t sin t

We already know:

sin 2t = 2 sin t cos t

and

cos 2t = cos²t − sin²t

Substitute them:

sin 3t = (2 sin t cos t)cos t + (cos²t − sin²t)sin t

Expand:

sin 3t = 2 sin t cos²t + sin t cos²t − sin³t

Combine the first two terms:

sin 3t = 3 sin t cos²t − sin³t

But we want a formula involving only sin t.

Use the Pythagorean identity:

cos²t = 1 − sin²t

Therefore:

sin 3t = 3 sin t(1 − sin²t) − sin³t

Expand:

sin 3t = 3 sin t − 3 sin³t − sin³t

Hence:

sin 3t = 3 sin t − 4 sin³t

So the familiar triple-angle formula has been derived entirely from the addition formula.

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4. Deriving cos 3t

Again:

3t = 2t + t

Therefore:

cos 3t = cos(2t + t)

Using the cosine addition formula:

cos 3t = cos 2t cos t − sin 2t sin t

Substitute the double-angle formulas:

cos 3t = (cos²t − sin²t)cos t − (2 sin t cos t)sin t

Expand:

cos 3t = cos³t − sin²t cos t − 2 sin²t cos t

Combine the last two terms:

cos 3t = cos³t − 3 sin²t cos t

Now use:

sin²t = 1 − cos²t

So:

cos 3t = cos³t − 3(1 − cos²t)cos t

Expand:

cos 3t = cos³t − 3 cos t + 3 cos³t

Therefore:

cos 3t = 4 cos³t − 3 cos t

Again, there was no separate rule for 3t.

We simply used:

3t = 2t + t

and the angle-addition formula.

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5. Now derive cos 4t

The same method works for 4t.

Write:

4t = 3t + t

Therefore:

cos 4t = cos(3t + t)

Using the cosine addition formula:

cos 4t = cos 3t cos t − sin 3t sin t

Now substitute the formulas we already derived:

cos 3t = 4 cos³t − 3 cos t

and

sin 3t = 3 sin t − 4 sin³t

Therefore:

cos 4t = (4 cos³t − 3 cos t)cos t − (3 sin t − 4 sin³t)sin t

Expand:

cos 4t = 4 cos⁴t − 3 cos²t − 3 sin²t + 4 sin⁴t

Since

sin²t + cos²t = 1

we can write:

cos 4t = 4 cos⁴t + 4 sin⁴t − 3

Now replace sin²t with 1 − cos²t:

sin⁴t = (1 − cos²t)²

Therefore:

cos 4t = 4 cos⁴t + 4(1 − cos²t)² − 3

Expand:

cos 4t = 4 cos⁴t + 4(1 − 2cos²t + cos⁴t) − 3

So:

cos 4t = 4 cos⁴t + 4 − 8cos²t + 4cos⁴t − 3

Finally:

cos 4t = 8 cos⁴t − 8 cos²t + 1

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6. Deriving sin 4t

Once again:

4t = 3t + t

Therefore:

sin 4t = sin(3t + t)

Using the sine addition formula:

sin 4t = sin 3t cos t + cos 3t sin t

Substitute:

sin 3t = 3 sin t − 4 sin³t

and

cos 3t = 4 cos³t − 3 cos t

Therefore:

sin 4t = (3 sin t − 4 sin³t)cos t + (4 cos³t − 3 cos t)sin t

After simplifying:

sin 4t = 4 sin t cos³t − 4 sin³t cos t

Factor:

sin 4t = 4 sin t cos t(cos²t − sin²t)

Since

cos²t − sin²t = cos 2t

and

2 sin t cos t = sin 2t

we can also write:

sin 4t = 2 sin 2t cos 2t

This is actually the double-angle formula applied to the angle 2t.

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7. What about 5t?

Nothing fundamentally new happens.

We simply write:

5t = 4t + t

Then:

sin 5t = sin(4t + t)

and

cos 5t = cos(4t + t)

Using the addition formulas:

sin 5t = sin 4t cos t + cos 4t sin t

and

cos 5t = cos 4t cos t − sin 4t sin t

We can substitute the formulas for sin 4t and cos 4t and simplify.

The resulting expressions become higher-degree polynomials in sin t or cos t.

For example:

sin 5t = 5 sin t − 20 sin³t + 16 sin⁵t

and

cos 5t = 16 cos⁵t − 20 cos³t + 5 cos t

Again, these are not formulas we need to regard as independent facts. They can be generated from the same addition formulas.

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8. We can continue forever

The pattern is now clear:

2t = t + t

3t = 2t + t

4t = 3t + t

5t = 4t + t

6t = 5t + t

and so on.

So if we know the sine and cosine of nt, we can obtain the sine and cosine of (n + 1)t.

The general formulas are:

sin((n + 1)t) = sin(nt)cos t + cos(nt)sin t

and

cos((n + 1)t) = cos(nt)cos t − sin(nt)sin t

This is called a recurrence relationship: the formula for the next multiple is obtained from the previous one.

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9. There is a beautiful pattern behind all of this

Look at what happened:

To get 2t

We used:

t + t

To get 3t

We used:

2t + t

To get 4t

We used:

3t + t

To get 5t

We used:

4t + t

So we are repeatedly doing the same thing:

Take the angle we already have and rotate by another t.

This connects directly with the geometric interpretation of sine and cosine.

Remember that on the unit circle:

• cos t represents the horizontal coordinate.
• sin t represents the vertical coordinate.

When we add another angle t, we rotate the vector by t again.

The angle-addition formulas tell us exactly how the horizontal and vertical components change during that rotation.

That is why the same two formulas can generate an unlimited number of multiple-angle formulas.

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10. Why do the formulas become more complicated?

You may notice that:

sin 2t = 2 sin t cos t

is relatively simple.

But:

sin 3t = 3 sin t − 4 sin³t

is more complicated.

And:

sin 5t = 5 sin t − 20 sin³t + 16 sin⁵t

is even more complicated.

This happens because every time we add another t, we substitute the previous expressions into the addition formulas and then expand and simplify.

So the algebra becomes progressively larger.

But the underlying idea never changes.

It is always:

previous angle + t

followed by the same two addition formulas.

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11. Do we really need to memorize all these formulas?

Usually, no.

It is useful to know some common ones, such as:

sin 2t = 2 sin t cos t

cos 2t = cos²t − sin²t

sin 3t = 3 sin t − 4 sin³t

cos 3t = 4 cos³t − 3 cos t

But the more important thing to understand is where they come from.

If you remember the two fundamental formulas

sin(A + B) = sin A cos B + cos A sin B

and

cos(A + B) = cos A cos B − sin A sin B

you can recreate the multiple-angle formulas whenever necessary.

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12. The big picture

We can therefore think of the entire process as a chain:

Angle-addition formulas

↓

Double-angle formulas

↓

Triple-angle formulas

↓

Four-angle formulas

↓

Five-angle formulas

↓

Six-angle formulas

↓

and so on…

There is no separate mathematical trick for each one.

They are all generated from the same fundamental principle:

Multiple angles are repeated additions of the same angle, and the angle-addition formulas tell us how sine and cosine behave when angles are added.

This is why understanding the angle-addition formulas is much more valuable than simply memorizing a long list of multiple-angle formulas.