How to memorize the trigonometric identities?
I am stuck trying to memorize the trig identities, and try as I may, I just can't get them to stick (especially the sum-product and product-sum formulas). I am concerned I won't be able to memorize them in time for my test, and I was wondering if there was a better way than rhote memorization.
Any suggestions?
Thanks.
EDIT: Passed the test, thanks to your good suggestions. Thank you all!
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$\begingroup$To be honest, I would always forget all my trig identities until I learned complex numbers. Assuming you don't want to go there, try to be as efficient as possible. I tell my students to remember these at the bare minimum:
$$ \cos(A\pm B) = \cos(A)\cos(B)\mp \sin(A)\sin(B)\\ \sin(A\pm B) = \sin(A)\cos(B)\pm \cos(A)\sin(B) $$
Where `$\mp$' means to flip the sign i.e. $A+B$ inside becomes $-$ outside. Then, you can get a lot of the other identities by simply adding or subtracting these! For example, the product-to-sum rule for cosine comes from adding the formulae for $\cos(A+B)$ and $\cos(A-B)$, and the product-to-sum rule for sin comes by subtracting them.
In an exam setting, you'll always be more efficient by having things memorized though, so going `from scratch' should probably be a last resort.
$\endgroup$ 1 $\begingroup$If you're familiar with basic manipulation of complex numbers, one way to remember (or quickly "re-derive") various identities is to use DeMoivre's formula:
$$e^{i\theta} = \cos\theta+i\sin\theta.$$
For instance, you can use this to find a formula for $\sin 2 \theta$ as follows:
$$\sin 2\theta = Im(e^{i(2\theta)}) = Im((e^{i\theta})^2) = Im((\cos \theta+i\sin\theta)^2)=$$
$$Im(cos^2\theta-\sin^2\theta+2i \cos\theta\sin\theta).$$
Taking the imaginary part of this last expression, we see that
$$\sin 2\theta=2\cos\theta\sin\theta.$$
$\endgroup$ 2 $\begingroup$From $$e^{i\theta}=\cos\theta+i\sin\theta$$ and the property of exponential you can find all the trigonometric relation you want.
$\endgroup$ 3 $\begingroup$The way that I used to memorize the sum of angle formulas is by memorizing the double angle formulas instead. For example $$\sin 2\theta = 2\sin\theta\cos\theta = \sin\theta\cos\theta+\cos\theta\sin\theta$$ reminds me that $$\sin(A+B) = \sin A\cos B + \cos A\sin B$$ similarly $$\cos2\theta = \cos^2\theta-\sin^2\theta = \cos\theta\cos\theta - \sin\theta\sin\theta$$ reminds me that $$\cos(A+B) = \cos A\cos B - \sin A\sin B$$
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