Question

precalculus question: how do you know that you've found all the values of x when asked to find all values of x satisfying (for example) 2(cosx)^2-cosx=0 on the interval (0, 2pi)

Answer 1

this is a quadratic equation that can be simplified.

cos(x)(2cos(x) - 1)=0

so cos(x) = 0 and 1/2

now you with need to know about the cos(x) curve or the quadratic...

cos is positive in the 1st and 4th quadrants...

here is a rough sketch of the y=cos(x)



use an exact value triangle to find cos(x) = 1/2

1st quadrant is just the angle, 4th quadrant is 2pi - angle.

hope it helps