Question

If 2cos²(π+x)+3sin(π+x) vanishes then the values of x lying in the interval from 0 to 2π is x=π/6 or 5π/6. Explain how?

Answer 1

We have to prove that `(pi/6)` and `(5pi)/6` are the 2 roots of the equation:
`2cos^2 (pi + x) + 2sin (pi + x) = 0`
First, replace in the equation x by `pi/6, or x = 30^@`, then, evaluate the equation.
`2cos^2 (pi + pi/6) = 2cos^2 ((7pi)/6) = 2cos^2 (210)) = 1.50`
`3sin ((7pi)/6) = 3sin (210) = 3(-0.5) = - 1.50`. Proved.
Next, replace x by `(5pi)/6, or x = 150^@`:
`2cos^2 ((11pi)/6) = 2cos^2 (330) = 2(0.75) = 1.50`
`3sin ((11pi)/6) = 3sin (330) = 3(-0.5) = - 1.50`. Proved