Triganometry(cosine of an angle)

Discussion in 'Math' started by chunkmartinez, Jan 21, 2007.

  1. chunkmartinez

    Thread Starter Senior Member

    Jan 6, 2007
    180
    1
    Can some one tell how to find the cosine of an angle with only the angle and the
    hypotenuse.
     
  2. Papabravo

    Expert

    Feb 24, 2006
    9,905
    1,723
    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.
     
  3. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    143
    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
     
  4. Papabravo

    Expert

    Feb 24, 2006
    9,905
    1,723
    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:

    Code ( (Unknown Language)):
    1.  
    2. cos x = 1 + a2x^2 + a4x^4 + a6x^6 + a8x^8 + a10x^10 + e(x)
    3.  
    4. 0 .le. x .le. pi/2
    5. norm(e(x)) .le. 2e-9
    6.  
    7. a2  = -.49999 99963
    8. a4  = +.04166 66418
    9. a6  = -.00138 88397
    10. a8  = +.00002 47609
    11. a10 = -.00000 02605
    12.  
    which would be adequate for single precision with 8 significant digits.
     
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