Chapter 5.4 - The Sum and Difference Formulas

Today in class we discussed and derived the sum and difference formulas for the three main trigonometric functions (sine, cosine, and tangent).

Before we jump right into the actual deriving of the formulas, let's review what the formulas actually are.

Sine:

Cosine:

Note: While the addition/subtraction signs correspond in the sine functions, they aren't the same for cosine.

Tangent: 

Now we will derive the above formulas (not all of them) so we can see where they come from and why they are the way that they are. 

Deriving the Sine Formula

First, draw the following picture.

(Angle DFC is a right angle but it is missing the little box indicating a right angle)

What we are trying to do is create a formula that allows us to easily find the sine the angle at point A, which is x + y. 

Now, think about the first step that we would take to find the sine of (x + y). 

Sine is opposite over hypotenuse. The opposite side is leg DE and the hypotenuse is AD. 

Therefore,

Based on the length adding postulate, DE can become DF + FE. FE is the same as CB, so we can substitute that which gets you:

Now, it gets tricky. DF and CB mean absolutely nothing to us if we're thinking in trig form, so now we have to switch our brains to see these side lengths as trigonometric functions.  

Refer back to the initial triangle: 

(Angle DFC is a right angle but it is missing the little box indicating a right angle)

According to ∆CDF : the cosx = (DF / DC) 

Multiply each side by DC and you have DCcosx = DF which is part of the equation we're looking to replace. 

The same goes for ∆ABC : sinx = (CB / AC) which simplifies to ACsinx = CB

Now, the equation is starting to look a little bit more familiar:

We do the exact same steps to get rid of the DC and the AC in the numerator now. 

∆ACD : siny =  (DC / AD) so ADsiny = DC

and

∆ACD : cosy = (AC / AD) so ADsiny = AC

Substitute those in:

Divide out the 'AD's:

And there you go, one of the two sine sum and difference formulas. 

Following the same path, we can deduce that:

Deriving the Tangent Formula:

Don't worry, no more triangles for now. 

We know based on previous knowledge that tangent of any angle is the same as the sine of that angle divided by its cosine. 

There fore, we can conclude that:

Now, it's down to some basic algebraic principles of substitution and simplifying to find the easy tangent formula. 

By substituting into the addition formula we get:

To simplify this into all tangents, rather than a jumble of sines and cosines, we will multiply each side by one over cosxcosy as so:

Now, we have all sines and cosines without a jumble of different angles multiplied by each other. All we do is simplify into tangents and we have the completed formula:

Following the same path we can conclude that:

How to use the formulas:

These formulas can be used to find a trigonometric function that is not labelled on the unit circle.
Take π/12 for example, it's not a point on the unit circle, but we want to find out the sine value. In order to do this, one must find two values that add up or have a difference of π/12. 

Follow these steps:

Note: The answer to the last example should have a denominator of 4, not 2. (Sorry for the mistake).