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Precalculus Mathematics Homework Help
Today, in our class, we received a trigonometric equation
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[QUOTE="Ray Vickson, post: 6019412, member: 330118"] Here is one way that will work, but be prepared for pages and pages of algebra---or better yet, use a computer algebra package, as I did: I used Maple. You can use Euler's formulas (see [URL='https://en.wikipedia.org/wiki/Euler%27s_formula']https://en.wikipedia.org/wiki/Euler's_formula[/URL]) to write $$\cos(x) = \frac{1}{2} (u + v), \;\; \sin(x) = \frac{1}{2i} (u - v), $$ where $$u = e^{ix}, \;\; v = e^{-ix} = \frac 1 u.$$ Also, ##\cos(2x) = (1/2)(e^{2ix} + e^{-2ix}) = (1/2)(u^2 + v^2).## That gives your function $$f(x) = \sin^{10} (x) + \cos^{10} (x) - \frac{29}{16} \cos^4 (2x)$$ as $$f = -\left( \frac u 2 - \frac v 2 \right)^{10} + \left( \frac u 2 + \frac v 2 \right)^{10} - \frac{29}{16} \left( \frac{u^2}{2} + \frac{v^2}{2} \right)^4.$$ Expand it all out, set ##v = 1/u##, then simplify. You will get an expression of the form $$f = -\frac{P(u^4)}{32 u^8}, $$ where ##P(y)## is a 4th degree polynomial in ##y## with positive integer coefficients. Furthermore, ##P(y)## factors into two quadratics with integer coefficients, so the equation ##f(x) = 0## translates into the easily solvable equation ##P(y) = 0##, where ##y = u^4 = e^{4ix}.## [/QUOTE]
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Precalculus Mathematics Homework Help
Today, in our class, we received a trigonometric equation
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