Trig Equations with Multiple Trig Functions

**~berserk** · April 6th 2008, 03:37 PM

solve for Θ using the interval 0≤Θ≤360

2cos^2 Θ+sinΘ=1

The number of degrees in the smallest positive angle that satisfies the equation

3cos2x+2 sin x+1=0

**o_O** · April 6th 2008, 03:48 PM

$2\cos^{2} \theta + \sin \theta = 1$
$2 \left(1 - \sin^{2} \theta \right) + \sin \theta = 1$
$2 - 2\sin^{2} \theta + \sin \theta = 1$
$0 = 2\sin^{2} \theta - sin \theta - 1$

Factor and solve for $\sin \theta$ and then $\theta$

$3\cos (2x) + 2 \sin x + 1 = 0$
$3 \left( 1 - 2 \sin^{2} x \right) + 2 \sin x + 1 = 0 \quad \quad \mbox{Since: } \cos 2x = 1 - 2\sin^{2} x$
$3 - 6 \sin^{2} x + 2\sin x + 1 = 0$
$0 = 6\sin^{2} x - 2 \sin x - 4$
$0 = 3\sin^{2} x - \sin x - 2$

Again factor and solve.

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