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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.
Associated Topics:
High School Trigonometry

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