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

1 Answer
Nghi N.
Feb 17, 2016

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)
Answerspi/2, (7pi)/6 and (11pi)/6

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