Basic math review

Here we aim to go over some math things that will be useful in the course.  This is by no means exhaustive, and some of the things below may not come up.  But we hope this will be a useful place to turn as a first quick resource if you're feeling a little rusty . . .

Fraction division
To divide one fraction by another, remember: Kentucky Chicken Fried (Keep Change Flip):

$$\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times\frac{d}{c}$$

Geometry
Area of a triangle =  $$\frac {1}{2}\rm{base}\times\rm{height}$$
Area of a circle = $$\pi r^2$$
Circumference = $$2\pi r$$
Volume of a sphere = $$\frac{4}{3}\pi r^3$$

Differential volume in spherical coordinates $$dV=r^2 dr \sin \theta d\theta d\phi$$
If the function you are integrating against this differential volume is angle-independent, you can use
$$dV=4\pi r^2 dr$$

Finally, notice that if you integrate the above over r from 0 to R, you get
$$\int dV = 4\pi \int_)^R r^2 dr=\frac {4\pi}{3}R^3$$
---the volume of a sphere!  And if you ever forget what dV is, just take the expression for the volume of a sphere and differentiate to get dV for a spherically symmetric problem.

Trigonometry
Just comes from the Greek for "triangle measuring": so any time a triangle shows up in a problem, you probably want to think trig.

Mnemonic for sine, cosine, tangent: SOHCAHTOA, or "Some Old Horse Caught A Horse Taking Oats Away"---meaning,

Sine = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent

and recall that 1/Sine = Cosecant (abbreviated Csc)
and . . .           1/Cosine = Secant (abbreviated Sec)
Finally, . . .     1/Tangent = Cotangent (abbreviated Cot)

More advanced trigonometry
Euler's formula: $$e^{i\theta}=\cos\theta+i\sin\theta$$ with $$i=\sqrt{-1}$$

This formula can be used to write
$$\sin \theta =\frac{1}{2i}\big[e^{i\theta}-e^{-i\theta}\big]$$
and
$$\cos \theta =\frac{1}{2}\big[e^{i\theta}+e^{-i\theta}\big]$$

Notice that if you flip the sign of the angle, sine changes sign (punny?), while cosine does not---that is no surprise, because we know that sine is an odd function and cosine is an even one.

Euler's formula is great because you can also use it to quickly derive the double-angle formulae for sine and cosine:

$$(\sin \theta)^2=\frac{1}{2i}\big[e^{i\theta}-e^{-i\theta}\big]^2$$
$$=-\frac{1}{4}\big[e^{i2\theta}+e^{-i2\theta}-2\big]$$
$$=\frac{1}{2}\big[1-\cos2\theta \big]$$

Working through similar math for cosine:
$$(\cos \theta)^2=\frac{1}{2}\big[e^{i\theta}+e^{-i\theta}\big]^2$$
$$=\frac{1}{4}\big[e^{i2\theta}+e^{-i2\theta}+2\big]$$
$$=\frac{1}{2}\big[\cos2\theta+1\big]$$

Now notice that adding these results together gives
$$\sin^2\theta+\cos^2\theta=1$$
---hopefully a familiar identity!  It's really just the Pythagorean theorem applied to a triangle with hypotenuse 1, because by the definitions of sine and cosine, the legs of the triangle will be sine and cosine if the hypotenuse is 1.

Finally, note that you can even use Euler's formula to get formulae relating higher powers of sine and cosine to higher multiples of the angle, e.g. $$\sin^3\theta$$ in terms of $$\sin 3\theta,$$ for instance.  You would just use the Binomial Theorem to expand the binomial (two terms added together)
$$\big[e^{i\theta}\pm e^{-i\theta}\big]^n$$ out, and then simplify, just like we did for $$n=2$$ above.

Most useful approximation ever
90% of astrophysics problems involving two terms taken to a power can be handled using what's below.
Suppose you have something of the form
$$\big[1+\epsilon\big]^n,$$
where $$\epsilon\ll 1.$$
Using the Binomial Theorem to expand this out, you'd get
$$1+n\epsilon +\rm{tems\;in\;higher\;powers\;of\;\epsilon}.$$
Because $$\epsilon\ll 1,$$
all these higher powers can just be neglected, since taking anything less than 1 to a power makes it smaller, the higher the power the smaller.
So we have
$$\big[1+\epsilon\big]^n\approx 1+n\epsilon.$$
Note that in astrophysical problems you often may have to factor out something to get a 1 in the first term, and can then apply this expansion. For this reason it's always good to consider the relative scales of any two quantities that have the same dimensions.