Without the aid of a table or a calculator. How can I find them? Now, I don't want a method, I want a formula. I haven't been able to find one. Even if it's insanely long and confusing, I want a formula to find these values, or at least one. Please, tell me how it can be calculated.
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I used to wonder this myself. I can add, subtract, multiply, divide by hand, but how does a calculator calculate sine and cosine? Is there a little man in there drawing triangles?
One way to do it is to use the Taylor series at x = 0 (you can learn more about Taylor Series in calculus). Calculating sine and cosine in radians (1 radian = 180/pi degrees, so you can do the conversion if you need to)
sin x = x - x^3 / (3 * 2 * 1) + x^5 / (5 * 4 * 3 * 2 * 1) - x^ 7 / (7 * 6 * 5 * 4 * 3 * 2 * 1) + x^9 / (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) - ...
This is an infinite series, with each term being + or - and then using odd powers of x. This can also be written
sin x = x - x^3 / 3! + x^5 / 5! - x^7/7! + x^9 / 9! - ...
For cos x, we have a similar formula
cos x = 1 - x^2 / (2 * 1) + x^4 / (4 * 3 * 2 * 1) - x^6 / (6 * 5 * 4 * 3 * 2 * 1) + ...
Same idea. Alternating signs, but even powers of x, also written
cos x = 1 - x^2 / 2! + x^4 / 4! - x^6/ 6!
To calculate tangent, tan(x) = sin(x) / cos(x)
Both of these formulas ultimately converge for all values of x, but it is best to use the fact that
sin(x) = sin (x + 2 pi) and move the value of x between -pi and pi. It converges much faster this way.
One way to do it is to use the Taylor series at x = 0 (you can learn more about Taylor Series in calculus). Calculating sine and cosine in radians (1 radian = 180/pi degrees, so you can do the conversion if you need to)
sin x = x - x^3 / (3 * 2 * 1) + x^5 / (5 * 4 * 3 * 2 * 1) - x^ 7 / (7 * 6 * 5 * 4 * 3 * 2 * 1) + x^9 / (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) - ...
This is an infinite series, with each term being + or - and then using odd powers of x. This can also be written
sin x = x - x^3 / 3! + x^5 / 5! - x^7/7! + x^9 / 9! - ...
For cos x, we have a similar formula
cos x = 1 - x^2 / (2 * 1) + x^4 / (4 * 3 * 2 * 1) - x^6 / (6 * 5 * 4 * 3 * 2 * 1) + ...
Same idea. Alternating signs, but even powers of x, also written
cos x = 1 - x^2 / 2! + x^4 / 4! - x^6/ 6!
To calculate tangent, tan(x) = sin(x) / cos(x)
Both of these formulas ultimately converge for all values of x, but it is best to use the fact that
sin(x) = sin (x + 2 pi) and move the value of x between -pi and pi. It converges much faster this way.