The trigonometric addition formulas can initially look like formulas that simply have to be memorized:

$\sin(a+b)=\sin a\cos b+\cos a\sin b$ $\cos(a+b)=\cos a\cos b-\sin a\sin b$ $\tan(a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b}$

But these formulas are not arbitrary. They can be understood from the basic definitions of sine, cosine and tangent.

If a student already understands:

$\sin a=\frac{p}{h}$ $\cos a=\frac{b}{h}$

and:

$\tan a=\frac{\sin a}{\cos a}$

then the addition formulas can be developed step by step.

1. Start with the basic right triangle

For an angle a, consider a right triangle.

The three important sides are:

Opposite side: p
Adjacent side: b
Hypotenuse: h

Therefore:

$\sin a=\frac{p}{h}$ $\cos a=\frac{b}{h}$

And since tangent is the ratio of sine to cosine:

$\tan a=\frac{\sin a}{\cos a}$

Substituting the definitions gives:

$\tan a=\frac{p/h}{b/h}=\frac{p}{b}$

So far, everything comes directly from the right triangle.

The interesting question is what happens when we have two angles, a and b, and want to find the trigonometric ratios of their sum.

2. What does an angle represent?

It helps to think of an angle as a direction.

Imagine a line of length 1 pointing at an angle a.

Because its length is 1, its horizontal and vertical components are particularly simple.

The horizontal component is:

$\cos a$

The vertical component is:

$\sin a$

So we can represent the direction using the pair:

$(\cos a,\sin a)$

This is also exactly what happens on a unit circle, where the radius is 1.

A point at angle a has coordinates:

$(\cos a,\sin a)$

This gives us a very useful interpretation:

Cosine tells us the horizontal component, while sine tells us the vertical component.

3. Now add another angle b

Suppose the original direction is a.

Now rotate that direction through another angle b.

The total angle becomes:

$a+b$

The new direction therefore has coordinates:

(cos(a + b), sin(a + b))

How can the new horizontal and vertical components be calculated from the original components?

The answer comes from examining what happens to each component when the direction is rotated through b.

4. Deriving sin(a + b)

Let’s first concentrate on the vertical component.

The original direction has two components.

The first component

The horizontal component has length:

$\cos a$

When this component is rotated through angle b, its vertical contribution is:

$\cos a\sin b$

Why?

Because for an angle b, sine represents the vertical component.

The second component

The original vertical component has length:

$\sin a$

After the rotation through b, its vertical contribution becomes:

$\sin a\cos b$

Why?

Because cosine represents the component that remains in the corresponding direction after the rotation.

Add the two vertical contributions

The total vertical component is therefore:

$\cos a\sin b+\sin a\cos b$

But the final vertical component is, by definition:

$\sin(a+b)$

Therefore:

$\boxed{\sin(a+b)=\sin a\cos b+\cos a\sin b}$

This is the sine addition formula.

5. Deriving cos(a + b)

Now let’s examine the horizontal component.

Again, there are two contributions.

The original horizontal component has length:

$\cos a$

After rotation through b, its horizontal contribution is:

$\cos a\cos b$

Now consider the original vertical component:

$\sin a$

When the direction is rotated counterclockwise, part of this component contributes in the negative horizontal direction.

Its magnitude is:

$\sin a\sin b$

Because this contribution is in the opposite horizontal direction, we subtract it.

Therefore the total horizontal component is:

$\cos a\cos b-\sin a\sin b$

But the final horizontal component is:

$\cos(a+b)$

Therefore:

$\boxed{\cos(a+b)=\cos a\cos b-\sin a\sin b}$

This is the cosine addition formula.

6. Why is there a plus sign for sine but a minus sign for cosine?

This is one of the most useful things to understand rather than memorize.

For sine, both component contributions point in the positive vertical direction:

$\sin(a+b)=\sin a\cos b+\cos a\sin b$

So they are added.

For cosine, one contribution points in the positive horizontal direction while the other contributes in the negative horizontal direction:

$\cos(a+b)=\cos a\cos b-\sin a\sin b$

So one is subtracted.

Therefore, the signs are not arbitrary.

They come from the directions of the horizontal and vertical components.

7. Deriving tan(a + b)

Now we can derive the tangent formula without starting from another geometric argument.

We already know:

$\tan x=\frac{\sin x}{\cos x}$

Replace x with a + b:

$\tan(a+b)=\frac{\sin(a+b)}{\cos(a+b)}$

Now substitute the two formulas we have just derived:

$\tan(a+b)=\frac{\sin a\cos b+\cos a\sin b}{\cos a\cos b-\sin a\sin b}$

Now divide both the numerator and denominator by:

$\cos a\cos b$

The expression becomes:

$\tan(a+b)=\frac{\frac{\sin a}{\cos a}+\frac{\sin b}{\cos b}}{1-\frac{\sin a}{\cos a}\frac{\sin b}{\cos b}}$

But we already know:

$\frac{\sin a}{\cos a}=\tan a$

and:

$\frac{\sin b}{\cos b}=\tan b$

Therefore:

$\boxed{\tan(a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b}}$

So the tangent addition formula follows directly from the sine and cosine addition formulas.

8. The three formulas are connected

The important lesson is that these are not three independent formulas.

They form a logical chain.

We begin with the basic triangle definitions:

$\sin a=\frac{p}{h}$ $\cos a=\frac{b}{h}$

Then we interpret sine and cosine as vertical and horizontal components.

That leads to:

$\sin(a+b)=\sin a\cos b+\cos a\sin b$

and:

$\cos(a+b)=\cos a\cos b-\sin a\sin b$

Finally, because:

$\tan x=\frac{\sin x}{\cos x}$

we obtain:

$\tan(a+b)=\frac{\tan a+\tan b}{1-\tan a\tan b}$

9. A numerical example

Let’s take:

$a=30^\circ$

and:

$b=45^\circ$

We know:

$\sin30^\circ=\frac{1}{2}$ $\cos30^\circ=\frac{\sqrt3}{2}$ $\sin45^\circ=\frac{\sqrt2}{2}$ $\cos45^\circ=\frac{\sqrt2}{2}$

Using the sine addition formula:

$\sin(30^\circ+45^\circ)=\sin30^\circ\cos45^\circ+\cos30^\circ\sin45^\circ$

Therefore:

$\sin75^\circ=\frac12\frac{\sqrt2}{2}+\frac{\sqrt3}{2}\frac{\sqrt2}{2}$

which gives:

$\sin75^\circ=\frac{\sqrt2+\sqrt6}{4}$

The formula therefore allows us to calculate the sine of 75° using the familiar angles 30° and 45°.

10. The deeper idea

The most important concept is not the formulas themselves.

It is the idea of components.

A direction can be broken into:

a horizontal component
a vertical component

Cosine describes the horizontal component.

Sine describes the vertical component.

When the direction is rotated by another angle, those components change. The addition formulas describe exactly how they change.

This is why the formulas have the structure:

$\sin(a+b)=\sin a\cos b+\cos a\sin b$

and:

$\cos(a+b)=\cos a\cos b-\sin a\sin b$

They are describing how the original horizontal and vertical components combine after a rotation.

11. From triangle ratios to a more powerful mathematical idea

This is also an important transition in learning mathematics.

At first, sine and cosine can be introduced simply as ratios in a right triangle:

$\sin a=\frac{\text{opposite}}{\text{hypotenuse}}$ $\cos a=\frac{\text{adjacent}}{\text{hypotenuse}}$

But later, the same functions become much more powerful.

They can describe:

coordinates on a circle
horizontal and vertical components
rotations
waves and oscillations
periodic motion
vectors
physics
engineering
signal processing

So the addition formulas are an excellent example of how a simple idea learned from a right triangle develops into a much broader mathematical tool.

The Formulas to Remember

After understanding the derivation, the three formulas become much easier to remember.

Sine addition

$\boxed{\sin(a+b)=\sin a\cos b+\cos a\sin b}$

Cosine addition

$\boxed{\cos(a+b)=\cos a\cos b-\sin a\sin b}$

Tangent addition

$(\tan a+\tan b)/(1-\tan a\tan b)$

The goal is not simply to memorize these formulas.

The better goal is to understand where they come from: starting with right-triangle ratios, interpreting sine and cosine as components, examining what happens when a direction is rotated, and finally using the relationship between tangent, sine and cosine.

How Are the sin(a + b), cos(a + b) and tan(a + b) Formulas Derived?

1. Start with the basic right triangle

2. What does an angle represent?

3. Now add another angle b