In Geometry, we were using sine, cosine, and tangent to find different angles and sides of the triangle, but my teacher didn't explain what they really are. Basically I just did what he told me to do without really understanding what they actually are for.
Can someone please explain to me what they really mean and what they do?
Well, I am really interested to provide the answer since same problem encountered me when I was around your age. Teachers mostly don't tell us what they are and what concepts are underlying them (sine and cosine). I was exhausted with using tricks to memorize where, when and how to use them. But later I found what is underlying them or from where are they derived? If once you come to know what are they really, they will be like your toys and you could play with them.
To understand what sine, cosine and tangent are you have to understand the term "function" in criteria of Mathematics.
I would like to provide the definition of rule form of function because I think it is an intuitive one.
Definition:
A function is a rule that produces a correspondence between two sets of elements such that to each element in the first set there corresponds one and only one element in the second set
The first set is called domain and the second one is called range.
It is usually denoted by the symbol $f$
Think function as a machine in which we input some data, it executes the process and give it to us in the form of output.
For Instance:
let $f(x)=2x$
then, $f(0)=0$,
$f(i)=2i$,
$f(3)=6$
Don't think that $f$ is some constant and being multiplied by $x$ but it is the notation. May be you would have problem with the notation as Richard Feynman had but later he agreed to use the standard notations and so you should do otherwise you would have to face bigger problems in future. So, What have you noticed thus far? Can you see that we are plugging numbers in function and getting different output every time. So this is an intuitive approach to functions.
Now, finally, we shall draw our attention towards the origin of angular functions i.e. from where they came.
Trigonometric or Angular Functions:
There are basically two circular functions namely, sine and cosine. Others are ratios in terms of both of them or either of them ($\tan {\theta}=\frac{\sin{\theta}}{\cos{\theta}}$)
We choose a 2-D coordinate system so that this general angle ($\angle POM$) in above figure) is in standard position. In figure a unit circle (circle of radius 1) is drawn with center at the origin O. The terminal ray (${PO}$) of the angle cuts the circle at $P(x,y)$. Thus to every real number $\theta$, there corresponds a unique point $P(x,y)$
So the set of ordered pairs $[\theta ,(x,y)]$ defines a function with,
$domain=({\theta | \theta \in \mathbb R})$
and,
$range=[(x,y) | x^2+y^2=1, x,y \in \mathbb R]$
So,
$p(\theta)=(x,y)$ where $\theta \in \mathbb R$
i.e., $[(x,y)|x^2+y^2=1, x, y \in \mathbb R]$
We define $\sin[p(\theta)]$ by,
$\sin{\theta}=y$
and similarly,
$\cos{\theta}=x$
These functions are called trigonometric or circular or angular functions.
In general if the 2nd figure not contains a unit circle, then we define
$\sin{\theta}=\frac{y}{\overline{OP}}$
and $\cos{\theta}=\frac{x}{\overline{OP}}$
They are all ratios of different sides of triangles. Have you learned "SOH-CAH-TOA"? Sine is the ratio of the opposite side over the hypotenuse, cosine is the ratio of the adjacent side over the hypotenuse, and tangent is the ratio of the opposite side over the adjacent side. For example, $\sin{30}=\frac{1}{2}$. This means that for every right triangle with a 30 degree angle, the opposite side over the hypotenuse is always going to be $\frac{1}{2}$, no matter what the size is.
Hope this helps.
Given a right triangle which has an angle $\theta$, we define $\sin(\theta)$ to be the ratio of the length of the side opposite $\theta$ to the length of the hypotenuse of the triangle. The beauty of this definition is that it does not depend on which right triangle you pick; as long as one of the angles is $\theta$, the ratio is the same as for any other right triangle with this angle, hence $\sin(\theta)$ is well-defined.
We define $\cos(\theta)$ and $\tan(\theta)$ to be the other familiar ratios of side lengths of a right triangle with an angle $\theta$.
