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
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$\begingroup$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.
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