I haven't taken this in school yet but I love math and physics and join lots of competitions. I have a problem with sine, cosine, and tangent, that I really need a SIMPLE explanation along with an example. I know for a first that they are related to angles and are used in physics, so how are they calculated and how are they used?
Thanks in advance
It is all about triangles and relations between the side lengths and angle amplitudes.
In a right triangle, by definition,
$$\cos(\theta)=\frac AC$$
$$\sin(\theta)=\frac BC$$
$$\tan(\theta)=\frac BA$$
Using these three interrelated functions, you can solve a real lot of geometric problems, such as relations between angles and sides of a general triangle, or even a triangle drawn on a sphere, and do topography, mechanics, geography, GPS, optics, astronomy... and more.
It turns out that these functions also have a deep meaning in maths and they are about as essential as the four basic operations and exponentiation. But this is an advanced topic.
To compute them, calculators have built-in algorithms, mostly based on approximation polynomials. E.g., for an angle expressed in radians,
$$\cos(\theta)\approx1-\frac{\theta^2}2+\frac{\theta^4}{4!}-\frac{\theta^6}{6!}+\cdots$$
and the more terms you add, the closer you get to the exact value.
Amazingly, to solve the apparently innocuous equation
$$4x^3-3x=\frac13,$$ you need the trigonometric functions.