# What is the purpose of the compound angle identity in trigonometry?

This may be a silly question, but one that I am confused about nonetheless.

With regards to the compound trig identities such as $\cos(A+B)=\cos A\cos B - \sin A\sin B$ etc., I'd like to know why they are used. What's the purpose? Surely, one would ask themselves that if we can just add $A$ and $B$ together and call it $C$ (degrees or radians) then we could just find cosine of $C$...?

I know there's more to it, but that's where I'd love to get your help.

Thank you! :)

• Without this property, the only way you can find the sine or cosine of an angle is by drawing the right triangle and making a measurement, which gives you at best an approximation. If you want the exact value, you need one of these formulas, or one of the many consequences thereof. May 13, 2015 at 20:10
• What you should be asking yourself, then, is how do we "just find cosine of $C$", if not with the help of these formulas? May 13, 2015 at 20:20
• What if A and B are variables and don't have known values? May 13, 2015 at 20:32

Consider finding the derivative of $f(x)=\cos(x)$.

With that identity the $f(x+h)=\cos(x+h)$ can be expanded and handled well whereas without it, how would you derive this value other than by defining it that way?

Using the identity, then things can be evaluated and computed.

Additionally, people commonly just memorize cosine and sine values for $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ$.

Using this theorem (and others) however, we can exactly determine several more values for sine and cosine that were previously inaccessible to pen&paper approaches. For example, $\cos 75^\circ = \cos(45^\circ+30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2}\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2}\frac{1}{2} = \frac{\sqrt{6}-\sqrt{2}}{4}$

(certainly, technology nowadays is wonderful and can tell us answers to questions like these, but it is good to know how to do things before calculators were around)

Here's an example of where it can be useful. Let's use it on $\cos(x+\pi/2)$. The formula says $\cos(x+\pi/2)=\cos(x)\cos(\pi/2)-\sin(x)\sin(\pi/2)$ and since $\cos(\pi/2)=0$ and $\sin(\pi/2)=1$ this implies $\cos(x+\pi/2)=-\sin(x)$ The sum formulas can be used to prove all kinds of useful identities like that.

• That's a somewhat artificial example, because the right triangle definition easily gives us $\cos(\pi/2-x)=\sin(x)$, so $\cos(\pi/2+x)=\sin(-x)=-\sin(x)$.
– Ian
May 13, 2015 at 20:31
• Sure, in fact the identity is geometrically obvious if you look at the graphs. But it illustrates the point, there are a zillion identities, not all of them are visually apparent. May 13, 2015 at 20:34

They aren't really used in geometry itself, but more in solving geometric equations and when doing trig substitutions in calculus. Furthermore, it is just interesting to know these properties, even if we rarely use them.

The purpose of these equations are to enable a student, mathematician, engineer, etc... to solve equations. You will see it alot in the following classes: Integral Calculus, Multivariable and Vector Calculus, Differential Equations, etc....

A really good example that is important in signal processing is the Fourier Series. It's a form of interpolation which is used to build waveforms.

As someone else mentioned, you could use a calculator to do this, but for pencil and paper work without a calculator, these identities come in handy to solve such equations. There are many such identities in all of mathematics, but no one person can know them all, not even the one's who have PHD degrees. The ones they teach in school are the most popular/useful ones.

Such formulae are used in a whole variety of ways, for example:

(i) In analysing or describing rotations of the plane or of space

(ii) In analysing wavelike phenomena in physics - eg quite simply, combining two sound waves of slightly different frequencies gives rise to a phenomenon known as "beats". The compound angle identities tell us why

(iii) There are quite a lot of common integrals which rely on such identities, including those used in fourier analysis

(iv) in taking the derivative of the cosine, for example, analysing the cosine of a slightly different angle in terms of the original angle

All these examples are related to each other, and some will probably seem rather surprising. I believe such formulae were also used in the days before logarithms to do multiplications using the trigonometric tables available to navigators.