Volatile Yard 挥洒自如 Write anything I want to write...

We normally need to learn some important trigonometric identities in high-school. For the identities among the trigonometric functions, some are 'manageable' and are not difficult to memorize. E.g.,

• Reciprocal Identities (E.g., $\sin \alpha = \frac{1}{\csc \alpha}$)
• Pythagorean Identities (E.g., $\sin^2 \alpha + \cos^2 \alpha = 1$)
• Quotient Identities (E.g., $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$)

But some are not easily to memorize. In particular, I used to get confused with the following identities:

• Sum-Difference Formulas
• Sum-to-Product Formulas
• Product-to-Sum Formulas

Why is it so? I analyzed them and made the following observations:

• The former formulas can be easily derived from the definitions of the trigonometric functions, normally involve only one angle, can obtain them by 'intuition' with the help of a right triangle, or some rather straightforward methods.
• The latter formulas involve two angles and there's a barrier to obtain the 'intuition' of manipulating them.

In particular, all the latter formulas can in fact be derived easily once you have $\sin (\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$. Unfortunately, the identity makes some sense but not much. (Proving this is in fact tedious although not difficult.) I have not found any existing trick for helping us in memorizing them. Deriving all of them from this formula is feasible but not practical.

To solve the problem, I analyzed their patterns and 'invented' my own way. I came out with the following diagram:

    +   -
S  SC  CS
C  CC -SS
-> 2 /2 /2
<- /2

With this I have been able to memorize those formulas with ease. I can't guarantee that this method helps you, because the diagram can be as confusing as the formulas themselves! If you have a better way, please share! Read on if you are interested in my method.

Usage

Here, the formulas have been written in a way that can somehow 'match' diagram easier. I use boldface to highlight the parts you need to use for each type of the formulas. Try to decode how the diagram can be used. If you get it, you probably can use it or even come out with something better. If you don't get it, I can only give you some simple hints... S for sin; C for cos; -> and <- show the directions.

(a) Sum-Difference Formulas

    +   -
S  SC  CS
C  CC -SS
-> 2 /2 /2
<- /2

$\displaystyle \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

$\displaystyle \sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$

$\displaystyle \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$

$\displaystyle \cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$

(b) Sum-to-Product Formulas

    +   -
S  SC  CS
C  CC -SS
-> 2 /2 /2
<- /2

$\displaystyle \sin \alpha + \sin \beta = 2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}$

$\displaystyle \sin \alpha - \sin \beta = 2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}$

$\displaystyle \cos \alpha + \cos \beta = 2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}$

$\displaystyle \cos \alpha - \cos \beta = - 2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}$

(c) Product-to-Sum Formulas

    +   -
S  SC  CS
C  CC -SS
-> 2 /2 /2
<- /2

$\displaystyle \sin \alpha \cos \beta = \frac{1}{2} \{ \sin (\alpha+\beta) + \sin (\alpha-\beta) \}$

$\displaystyle \cos \alpha \sin \beta = \frac{1}{2} \{ \sin (\alpha+\beta) - \sin (\alpha-\beta) \}$

$\displaystyle \cos \alpha \cos \beta = \frac{1}{2} \{ \cos (\alpha+\beta) + \cos (\alpha-\beta) \}$

$\displaystyle - \sin \alpha \sin \beta = \frac{1}{2} \{ \cos (\alpha+\beta) - \cos (\alpha-\beta) \}$

Are you ok or more confused now?