Convert trig of angle in degrees to decimal value without a calculator Been forever since I did trig, I know how to use my calculator to do it, but I can't remember if there is a way to evaluate $\cos(x)$ without a calculator.  For example:
$\cos(30)$ evaluates to $0.866025$
How would one come up with that value without plugging it into a calculator?
 A: Note that $$\cos{30^{\circ}} = \cos{\frac{\pi}{6}} = \frac{\sqrt{3}}{2} \approx 0.866025$$
since we can convert from degrees into radians by using
$$1^{\circ} = \frac{\pi}{180}$$

A decent "mnemonic" to remeber some basic trig function values is the following table:
$$\begin{array}{lr} t & \sin{t} \\ 
0             & \frac{\sqrt{0}}{1} \\ 
\frac{\pi}{6} & \frac{\sqrt{1}}{1} \\
\frac{\pi}{4} & \frac{\sqrt{2}}{2} \\
\frac{\pi}{3} & \frac{\sqrt{3}}{2} \\
\frac{\pi}{2} & \frac{\sqrt{4}}{2} \end{array}$$
Then $\cos{t}$ can be found from the relation $\cos^2 t + \sin^2 t = 1$.
A: For certain "special angles", like $0$, $\frac\pi6$, $\frac\pi4$, $\frac\pi3$, and $\frac\pi2$, there are very simple expressions for the sine and cosine. For somewhat less special angles, like $\frac\pi5$ and $\frac\pi{10}$, there are somewhat less simple expressions. In the general case, you can always estimate using, for example, $\cos x = 1-\frac {x^2}{2!}+\frac{x^4}{4!}-\cdots$ and $\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$, or other series expansions.
Warning: The series expansions will only work properly for radian angle measures. In general, modern mathematicians, physicists, etc., only ever use radian measure. Degrees are yesterday (ancient Babylonian, to be precise).
Example: Suppose you want to calculate $\cos 5$. Well, $\cos 5 = \cos (2\pi-5)$, and $2\pi-5\approx6.28-5=1.28$. We see that this is close to $\pi/2\approx 1.57$, so we will use the Taylor expansion about $\pi/2$:
$$\cos x = -(x-\pi/2)+\frac 1{3!}(x-\pi/2)^3-\cdots.$$
Taking just the first term gives $\cos 5\approx 0.29$. My calculator gives $\cos 5 \approx 0.28$, so it's good enough for government work.
A: The series expansion exists if you really want to get into non-nice values.
$$\cos(x) = \sum\limits_{k=0}^\infty \frac{(-1)^k }{(2 k)!}x^{2 k}$$
A: A simple derivation is:
An equilateral triangle has all sides equal and all angles equal to $60^{\circ}$.  Therefore a triangle with angle $30^{\circ}$ (which is a bisected equilateral) has a hypotenuse of 1 and a short side of $\frac12$, so $\sin(30^{\circ})=\frac12$.  Since $\sin^2\theta + \cos^2\theta = 1$, 
$$\cos(30^{\circ}) = \sqrt{1-\frac14} = \frac{\sqrt{3}}{2}$$.
The advantage of this derivation is that you can, well, derive it (easily).  But, $30^{\circ}$ is obviously an easy angle.  For more general angles, different formula are used depending on the need, for example, Taylor series (as others have shown) and CORDIC approximations are common on, eg, microcontrollers (where you can't do the fast multiplications required for a Taylor series approach).
