Question

How do you find all values of x in the interval [0, 2pi] in the equation `cos^2theta -sin^2theta + sintheta= 0`?

Answer 1

`x = pi/2, (7pi)/6 and (11pi)/6`

Explanation:

Replace in the equation `cos^2 x` by `1 - sin^2 x` -->
`1 - sin^2 x - sin^2 x + sin x = 0`
`- 2sin^2 x + sin x + 1 = 0.`
Solve this quadratic equation in sin x.
Since a + b + c = 0, use shortcut. The 2 real roots are: sin x = 1 and `sin x = c/a = -1/2`.
a. sin x = 1 --> arc `x = pi/2, or 90^@`
b. `sin x = -1/2` --> `x = -pi/6, or (11pi)/6` (co-terminal)
and `x = -(5pi)/6`, or `(7pi)/6` (co-terminal)
Answers`pi/2, (7pi)/6 and (11pi)/6`