Thread: find solutions in eqautions in interval

1. find solutions in eqautions in interval

$
\cos\theta = -\frac{1}{2}, 0<= \theta < 2\pi
$

how do we find all solutions in that interval given. thanks

2. Originally Posted by jvignacio
$
\cos\theta = -\frac{1}{2}, 0<= \theta < 2\pi
$

how do we find all solutions in that interval given. thanks
Hello JV,

$\cos \theta=-\frac{1}{2}$

$\cos 60=\frac{1}{2}$

Cosine is negative in QII and QIII.

Reference angle in QII = 180 - 60 = 120 degrees or $\frac{2\pi}{3}$ radians

Reference angle in QIII = 180 + 60 = 240 degrees or $\frac{4\pi}{3}$ radians

$\theta=\left\{\frac{2\pi}{3}, \frac{4\pi}{3}\right\}$ over the interval $0 \leq \theta \leq 2\pi$
$\theta=\left\{120^{\circ}, 240^{\circ}\right\}$ over the interval $0^{\circ} \leq \theta \leq 360^{\circ}$