exact solutions

Question

Find all exact solutions on [0, 2π).(Enter your answers as a comma-separated list.)

2 sin(θ) = negative square root 3

2 cos2(t) + 3 cos(t) = −1

2 tan2(t) = −3 sec(t)

10 sin2(x) + 7 sin(x) + 1 = 0

Answer 1

To solve the first one, divide both sides by 2 and then find where sin(θ) = -(√3)/2 on the unit circle.

The others can all be solved as quadratic equations by making an appropriate substitution such as u = cos(t).

You may find the trig identity tan²(t) = sec²(t) - 1 helpful.