# If cos theta =2/3,find theta where 180°<theta<360° What is the theta?

May 6, 2018

$\implies \theta = {\left(311.81\right)}^{\circ}$

#### Explanation:

Here,

costheta =2/3 > 0=>I^(st) Quadrant or color(red)(IV^(th) Quadrant...to(A)

But,

180^circ < theta < 360^circ=>III^(nd)Quadran or color(red)(IV^(th)Quadrant to(B)

From color(red)((A) and(B),we can say that

270^circ < theta < 360^circtocolor(red)(IV^(th) Quadrant

Hence,

$\cos \theta = \frac{2}{3} \implies \theta = {360}^{\circ} - {\cos}^{-} 1 \left(\frac{2}{3}\right) = {360}^{\circ} - {\left(48.19\right)}^{\circ}$

$\implies \theta = {\left(311.81\right)}^{\circ}$

May 6, 2018

${180}^{\circ} < \theta < {360}^{\circ} ,$ means third or fourth quadrant. A positive cosine means first or fourth quadrant. So fourth quadrant:

$\theta = {360}^{\circ} - \textrm{A r c} \textrm{\cos} \left(\frac{2}{3}\right) \approx {311.8}^{\circ}$

May 6, 2018

$\theta = {311.8}^{\circ} \text{ to 1 dec. place}$

#### Explanation:

$\text{since "costheta>0" then "theta" is in the first or}$
$\text{fourth quadrant}$

$\text{given "180^@ < theta<360^@" we require "theta" in the}$
$\text{fourth quadrant}$

$\theta = {\cos}^{-} 1 \left(\frac{2}{3}\right) = {48.2}^{\circ} \leftarrow \textcolor{red}{\text{in first quadrant}}$

$\Rightarrow \theta = \left(360 - 48.2\right) = {311.8}^{\circ} \leftarrow \textcolor{red}{\text{in fourth quadrant}}$

May 6, 2018

$t = {311}^{\circ} 81$

#### Explanation:

$\cos t = \frac{2}{3}$
Calculator and unit circle give 2 solutions for t:
$t = \pm {48}^{\circ} 19$
In the interval (180, 360), the answer is:
$t = - {48}^{\circ} 19$,
or $t = {360}^{\circ} - {48}^{\circ} 19 = {311}^{\circ} 81$ (co-terminal to - 48.19)