Triganometry(cosine of an angle)

Papabravo

Joined Feb 24, 2006
13,560
Can some one tell how to find the cosine of an angle with only the angle and the
hypotenuse.
The cosine is a function of the angle alone. The hypoteneuse is extraneous information. Most calculators will perform the function on an angle expressed in degrees, radians, or grads.

You can also look up the cosine in a table, or you can use a slide rule. That's what we did before the first scientific calculators were available.

If you want to know how the calulator evaluates the cosine function it probably uses a truncated chebyshev polynomial.
 

Dave

Joined Nov 17, 2003
6,970
The cosine of an angle is defined as:

Cosθ = A/H

You know θ and H.

If you want to know how the calulator evaluates the cosine function it probably uses a truncated chebyshev polynomial.
I would have thought that a calculator would use a Taylor Series to calculate sine and cosine? (This approach would be easy to program). It would be interesting to know.

Dave
 

Papabravo

Joined Feb 24, 2006
13,560
The cosine of an angle is defined as:

Cosθ = A/H

You know θ and H.



I would have thought that a calculator would use a Taylor Series to calculate sine and cosine? (This approach would be easy to program). It would be interesting to know.

Dave
In the original problem statement the length of the adjacent side is not known. The givens are theta and the length of the hypoteneuse. If you know theta, the hypoteneuse is irrelevant.

The problem with the Taylor series can be seen from the remainder theorem. To get an appropriate number of digits takes a considerable number of terms. The Chebyshev polynomial also has a bounded error over the domain of the approximation. I forget the details but I think excellent approximations have powers of theta no higher than the 11th.

Edit:
Abramowitz and Stegun list the following:

Rich (BB code):
cos x = 1 + a2x^2 + a4x^4 + a6x^6 + a8x^8 + a10x^10 + e(x)
 
0 .le. x .le. pi/2
norm(e(x)) .le. 2e-9
 
a2  = -.49999 99963
a4  = +.04166 66418
a6  = -.00138 88397
a8  = +.00002 47609
a10 = -.00000 02605
which would be adequate for single precision with 8 significant digits.
 
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