anonymous
  • anonymous
if cosx-2cos2x=0 then x=?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
Solve for x: cos(x)-2 cos(2 x) = 0 Simplify the left hand side. Transform cos(x)-2 cos(2 x) into a polynomial with respect to cos(x) using cos(2 x) = 2 cos^2(x)-1: 2+cos(x)-4 cos^2(x) = 0 Write the quadratic equation in standard form. Divide both sides by -4: -1/2-(cos(x))/4+cos^2(x) = 0 Solve the quadratic equation by completing the square. Add 1/2 to both sides: cos^2(x)-(cos(x))/4 = 1/2 Take one half of the coefficient of cos(x) and square it, then add it to both sides. Add 1/64 to both sides: 1/64-(cos(x))/4+cos^2(x) = 33/64 Factor the left hand side. Write the left hand side as a square: (cos(x)-1/8)^2 = 33/64 Eliminate the exponent on the left hand side. Take the square root of both sides: cos(x)-1/8 = sqrt(33)/8 or cos(x)-1/8 = -sqrt(33)/8 Look at the first equation: Solve for cos(x). Add 1/8 to both sides: cos(x) = 1/8+sqrt(33)/8 or cos(x)-1/8 = -sqrt(33)/8 Eliminate the cosine from the left hand side. Take the inverse cosine of both sides: x = cos^(-1)(1/8+sqrt(33)/8)+2 pi n_1 for n_1 element Z or x = 2 pi n_2-cos^(-1)(1/8+sqrt(33)/8) for n_2 element Z or cos(x)-1/8 = -sqrt(33)/8 Look at the third equation: Solve for cos(x). Add 1/8 to both sides: x = cos^(-1)(1/8+sqrt(33)/8)+2 pi n_1 for n_1 element Z or x = 2 pi n_2-cos^(-1)(1/8+sqrt(33)/8) for n_2 element Z or cos(x) = 1/8-sqrt(33)/8 Eliminate the cosine from the left hand side. Take the inverse cosine of both sides: Answer: | | x = cos^(-1)(1/8+sqrt(33)/8)+2 pi n_1 for n_1 element Z or x = 2 pi n_2-cos^(-1)(1/8+sqrt(33)/8) for n_2 element Z or x = cos^(-1)(1/8-sqrt(33)/8)+2 pi n_3 for n_3 element Z or x = 2 pi n_4-cos^(-1)(1/8-sqrt(33)/8) for n_4 element Z

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