How do you find sin cos and tan values manually?

Answer 1

This is so much work that it is not worthwhile to do in practice, although the formulae themselves are actually quite simple. The basic method is to use a so-called "infinite series". The angle must be expressed in radians. If the angle is in degrees, multiply it by (pi/180), to get the equivalent angle in radians. Then, use the formula:

sin(x) = x - x3/3! + x5/5! - x7/7! + x9/9!...

The individual terms become smaller and smaller, quite quickly, so the idea is to continue adding more terms until you see that the terms become so small that you can ignore them (depending on the desired degree of accuracy). An expression like 5!, read "five factorial" or "the factorial of five", means to multiply all natural numbers up to five: 5! = 1 x 2 x 3 x 4 x 5.

Similarly,

cos(x) = 1 - x2/2! + x4/4! - x6/6! + x8/8!...

There is a more complicated formula for tan(x), or simply calculate as follows:

tan(x) = sin(x) / cos(x)

The formulae for sin(x) and cos(x) are derived from the Taylor expansion, explained in basic calculus books.