SOLUTION: How do you algebraically solve x(sinx) = cosx ?

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Question 390443: How do you algebraically solve x(sinx) = cosx ?
Found 2 solutions by stanbon, richard1234:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
How do you algebraically solve
x(sinx) = cosx ?
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Divide both sides by sin(x) to get:
x = cot(x)
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Graph the line y = x
Graph the curve cot(x)
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Find the point(s) of intersection:
graph%28400%2C300%2C-10%2C10%2C-10%2C10%2Cx%2C1%2Ftan%28x%29%29
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Cheers,
Stan H.

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Not very easy to solve this algebraically, since trigonometric functions are transcendental (meaning "not algebraic"), and most equations that combine transcendental and algebraic functions cannot be solved algebraically. You could approximate the solutions graphically, or use a technique called Newton's method, which uses some introductory calculus.

Newton's method approximates the zeros of a function via a recursive sequence. Move all terms to one side to get x+sin+x+-+cos+x+=+0, and let f%28x%29+=+x+sin+x+-+cos+x. Pick an initial guess and denote it x%5B0%5D. Then, for n+%3E=+0,

x%5Bn%2B1%5D+=+x%5Bn%5D+-+f%28x%5Bn%5D%29%2F%28f%3A%28x%5Bn%5D%29%29 (The colon represents an apostrophe, the parsing system doesn't accept f'(x[n])).

Note that f'(x[n]) is the derivative of the function at x[n].

Repeating this process, x%5Bn%5D converges to one of the zeros of f(x).