Question

if cosx-2cos2x=0 then x=?

Answer 1

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