For what angle measure(s) between 0 and 2pi is 2cos^2x - cosx=1? Use mathematics to explain how you determined your answers.

Answer 1

If I understand correctly, the equation is 2cos²x - cosx = 1.

My advice, look at it like a quadratic equation.

For example, let y = cosx,

Now it looks like 2y² - y = 1

Bring the 1 over to the left side.

2y² - y - 1 = 0

Use quadratic formula or other methods to solve for y

The solutions are y = 1 and y = -0.5

Therefore, after replacing y with cosx, you get cosx = 1 and cosx = -0.5

You can use cos^-1 to solve for x.

So cos^-1(1) = x and cos^-1(-0.5) = x

Keep in mind there will be more than one solution for each version.

For example cos(0) = 1 and cos(2π) = 1

Same for cos(2π/3) = -0.5 and cos(4π/3) = -0.5

Therefore x = 0, 2π/3, 4π/3, and 2π

Hope this helps.