Finding cosines, sines, and tangents
Favorites|Homepage
Subscriptions | sitemap
HOME > Mathematics > Finding cosines, sines, and tangents

Finding cosines, sines, and tangents

[From: ] [author: ] [Date: 11-05-10] [Hit: ]
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.sin x = x - x^3 / 3!......
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.

-
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.
1
keywords: and,cosines,Finding,sines,tangents,Finding cosines, sines, and tangents
New
Hot
© 2008-2010 http://www.science-mathematics.com . Program by zplan cms. Theme by wukong .