Meanings of Sine, Cosine, Tangent Whenever I have a question dealing with sine, cosine, and tangent, my teacher always says to use a calculator. I would like to know how you would solve these without just using a calculator, that way I understand what is actually being done.
Could someone explain to me how this would be done, and what sine, cosine, and tangent actually represent? 
 A: $\sin(x)$, $\cos(x)$ and $\tan(x)$ are defined by the ratios of specific sides of right angled triangles.

$$\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}},\quad\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}},\quad\tan(A)=\frac{\text{opposite}}{\text{adjacent}}$$
Interestingly, the ratios are fixed values, regardless of the scale of the triangle. With some angles, this can be easy to find. For example, a $1$, $1$, $\sqrt{2}$ triangle gives us $\sin(45^o)=\cos(45^0)=\frac{1}{\sqrt{2}}$ but with others, there is no neat representation if any other representation at all.
When a calculator spits out a string of numbers for the $\sin$ of an angle, it's using specific algorithms, which have been coded in. One such algorithm is the Taylor series. When calculators use these algorithms they only do it in radians (as far as I know). To convert an angle from degrees to radians, multiply it by $\frac{\pi}{180}$.
So, one way of finding the $\sin$ of an angle (in radians) is $$\sin(x)=x-\frac{x^3}{1\cdot2\cdot3}+\frac{x^5}{1\cdot2\cdot3\cdot4\cdot5}+\ldots$$
This series has infinitely many terms and will get closer to a value, the more terms you use. If a calculator did this series, it would find the first load of terms then stop and gives what it had found (which is close to the real value but an approximation). You $could$ do this by hand if you had a lot of time on your hands or were some sort of Luddite. I'd imagine it would get quite boring.
Hope I helped!
A: The trigonometric functions are just special kind of functions. Like, e.g., $2x+4$, except they are harder to compute numerically.
They are defined for all real numbers; the definition you are probably familiar with (the ratios of sides of right triangles) is used to define them for small arguments ($[0;\pi/2]$) which is then extended for other values.
The trig functions have many nice properties, which means that they pop up all over math, so knowing how to handle them is important.
As to how to compute them - well, there is no way around it, it's hard. Before calculators there were tables and you would use those. The tables were constructed by hand by very dedicated people, using various approximation formulas.
And math does not stop there.  Special functions is a whole class of hard-to-compute-but-important functions, and trigs are just a small sample of those.
A: Before there were calculators (yes, I'm old enough to remember...) people used slide rules for rough calculations (maybe three significant digits if you were lucky) and tables of trig functions for more accuracy.
Trust me: you don't want to use either of those methods.
But you definitely should make an effort to understand 
what the trig functions mean, and their basic properties.  I hope your teacher
is teaching you that. 
