# find solutions in eqautions in interval

• Nov 21st 2008, 02:49 AM
jvignacio
find solutions in eqautions in interval
$\displaystyle \cos\theta = -\frac{1}{2}, 0<= \theta < 2\pi$

how do we find all solutions in that interval given. thanks
• Nov 21st 2008, 10:23 AM
masters
Quote:

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

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

Hello JV,

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

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

Cosine is negative in QII and QIII.

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

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

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