When deriving the trigonometric addition formulas geometrically, it is helpful to begin with a simple configuration.

For example, we can choose A to be an acute angle and choose B so that A + B remains in the first quadrant.

This makes the diagram much easier to understand.

However, there is an important distinction:

We are simplifying the diagram, not restricting the formula.

The resulting addition formulas work for much more general values of A and B.

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1. Why choose a convenient configuration?

Suppose we begin with a unit radius making an acute angle A with the positive x-axis.

Its components are easy to identify:

• horizontal component = cos A
• vertical component = sin A

Both components point in the positive directions:

             P
            /|
           / |
          /  | sin A
         /   |
        /    |
       O─────H
          cos A

Now rotate the radius anticlockwise through B.

If A + B remains in the first quadrant, the final radius is also easy to draw:

             P
            /
           /
          / A+B
         /
        O────────→

There is no immediate need to deal with negative coordinates or components pointing downward or to the left.

This makes the geometric relationships easier to see.

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2. What are we actually trying to understand?

We are not trying to prove that the formula works only for acute angles.

Instead, we are using this simple diagram to discover the structure of the formula.

We start with the two components of the original radius:

cos A

and

sin A

Then we rotate the entire radius through B.

The two original components produce contributions to the final horizontal and vertical components.

For the horizontal direction:

cos A produces a rightward contribution:

cos A cos B

while sin A produces a leftward contribution:

−sin A sin B

Therefore:

final horizontal component

= cos A cos B − sin A sin B

Since the final radius is at angle A + B:

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

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3. The same idea gives the sine formula

The vertical contributions are:

From the original vertical component:

sin A cos B

From the original horizontal component:

cos A sin B

Both point upward in this particular configuration.

Therefore:

final vertical component

= sin A cos B + cos A sin B

Since the final radius is at angle A + B:

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

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4. What if the angles are not in the first quadrant?

This is where the distinction becomes important.

Suppose the final angle moves into another quadrant.

The diagram may become more complicated.

Some components may point:

• left instead of right
• down instead of up

Consequently, some coordinates become negative.

But this does not invalidate the formulas.

The Cartesian coordinate system already provides the sign convention:

• right = positive x
• left = negative x
• up = positive y
• down = negative y

So the same component reasoning continues to work; we simply have to keep track of directions.

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5. A simple example

Suppose A + B lies in the first quadrant.

The final point has:

positive horizontal coordinate

and

positive vertical coordinate

So:

cos(A + B) > 0

and:

sin(A + B) > 0

Now imagine that A + B moves into the second quadrant.

The final point is now to the left of the y-axis but still above the x-axis.

Therefore:

cos(A + B) < 0

while:

sin(A + B) > 0

Nothing about the formula itself has changed.

The coordinate system simply tells us that the horizontal component is now negative.

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6. The formula is more general than our diagram

This is an important principle in geometric derivations.

A diagram often uses a convenient special configuration so that the relationships can be seen clearly.

That does not mean the resulting mathematical relationship is restricted to that configuration.

In our case, we use an acute A and a suitable B because it makes the component geometry easier to visualize.

Once we obtain:

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

and

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

these identities hold for general real angles.

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7. The same applies to subtraction

The subtraction formulas can be obtained by taking:

B → −B

because:

A − B = A + (−B)

We then use:

cos(−B) = cos B

and:

sin(−B) = −sin B

This gives:

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

and:

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

Again, the original diagram may have been drawn using convenient acute angles, but the resulting formulas are not restricted to that situation.

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The Key Insight

There are two different things to keep separate:

The diagram

We choose a convenient configuration—such as A and A + B lying in the first quadrant—to make the geometry easy to follow.

The mathematical identity

The formula we discover from that geometry is much more general.

A simple diagram helps us see the relationship; it does not limit the range of the relationship.

So when looking at our derivation, don’t think:

“The formula works because A and A + B are in the first quadrant.”

Instead think:

“We chose the first-quadrant configuration because it makes the component relationships easy to see. Once those relationships are established, the resulting trigonometric identities apply generally.”

This distinction allows us to use an intuitive geometric picture without confusing the convenience of the diagram with the generality of the formula.