Sine, Simply?Date: 10/27/2011 at 02:31:59 From: Chris Subject: Formulas for sine, cosine and tangent Hi, I'm in High School Geometry, and my teacher has just started to go over trigonometric ratios. He's showed us how to type sin(theta), cos(theta) and tan(theta) into the calculator, but I was wondering if there was a formula for them. I've read about how calculators use simple math to solve complex algorithms, but most of what I've found went over my head, mainly because it didn't come with much explanation. For example, I found something about radians: Given x = radians sin x = x - x3/3! + x5/5! - x7/7! But I haven't been taught radians. And I found something else about calculation through a complex plane: "Calculators often use the CORDIC algorithm to find values of trigonometric functions, which is based on thinking of the angle as the phase of a complex number in the complex plane, and then rotating the complex number by multiplying it by a succession of constant values." That went way over my head. So I was wondering if you would be able to explain a simpler formula? For example, I tried to figure out the relationship between sine of an angle and its complement. I think logically, so understanding the work behind sin, cos, and tan would, I think, help me get it. P.S. My teacher has also taught us SohCahToa, and that they're ratios. But I'm not looking for how to apply them to triangles; I'm looking to find the link between x and sin(x). Date: 10/27/2011 at 08:41:10 From: Doctor Jerry Subject: Re: Formulas for sine, cosine and tangent Hello Chris, Thanks for writing to Dr. Math. The most straight-forward method for calculating sin(x) and cos(x), where x is in radians, is to use their series representations: sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ... cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ... These formulas are easily derived once you know some calculus. They are called "Maclaurin series." For small x, the first three terms of the above series give quite good accuracy, as shown in the third and fifth columns of this small table: x sin(x) x - x^3/3! + x^5/5! cos(x) 1 - x^2/2! + x^4/4! -------------------------------------------------------------------- 0.05 0.0499792 0.0499792 0.99875 0.99875 0.1 0.0998334 0.0998334 0.995004 0.995004 0.15 0.149438 0.149438 0.988771 0.988771 0.2 0.198669 0.198669 0.980067 0.980067 0.25 0.247404 0.247404 0.968912 0.968913 0.3 0.29552 0.29552 0.955336 0.955338 0.35 0.342898 0.342898 0.939373 0.939375 0.4 0.389418 0.389419 0.921061 0.921067 0.45 0.434966 0.434966 0.900447 0.900459 0.5 0.479426 0.479427 0.877583 0.877604 Calculators use more complicated expressions, such as CORDIC, for efficiency. Please feel free to write back -- using the URLs at the bottom of this message -- if you have questions relative to my comments. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 10/27/2011 at 09:43:26 From: Chris Subject: Formulas for sine, cosine and tangent Doctor Jerry, You wrote: "Calculators use more complicated expressions, such as CORDIC, for efficiency." What's a CORDIC expression/complex plane? Date: 10/27/2011 at 12:11:54 From: Doctor Jerry Subject: Re: Formulas for sine, cosine and tangent Hello Chris, I'm not familiar with the details of CORDIC. However, here's an answer I wrote a year or two ago: TI and HP use several stored constants and calculate the values of several sequences of numbers using only addition and multiplication. They continue the calculation until sufficient accuracy is obtained. The algorithm looks like this. I'll use x_k to mean x sub k, and x_{k + 1} to mean x sub k + 1, and so on. Let x_{k + 1} = x_k - d_k*y_k*2^(-k) y_{k + 1} = y_k + d_k*x_k*2^(-k) z_{k + 1} = z_k - d_k*s_k The numbers d_k are equal to the sign of z_k (if z_k >= 0, d_k = 1; if z_k < 0, then d_k = -1). Also, s_k = arctan(2^(-k)) The numbers s_k are permanently stored in the calculator, maybe up to k = 50 or so. Starting values for the calculation are calculated. If z_0 = t is given, where t is a given angle (in radians), then y_0 = 0 and x_0 = cos(s_0)*cos(s_1)*...*cos(s_{47}) As k increases, x_k approaches cos(t) and y_k approaches sin(t). You might try "googling" CORDIC. Please feel free to write back -- using the URLs at the bottom of this message -- if you have questions relative to my comments. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 10/28/2011 at 01:26:11 From: Chris Subject: Thank you (Formulas for sine, cosine and tangent) Just a thank you for helping me understand the math behind sin, cos, and tan. Though the algorithms went way over my head, now I get the gist of how they work. |
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