Question

Please answer #12 as seen in the picture. Thank you.

Answer 1

If we choose a=20, b=15 and c=11 ans substitute these values into the given formula, we have

$\cos O=\frac{15^2+11^2-20^2}{2(11)(15)}$

which gives

$\cos O=\frac{346-400}{330}$

then

$\begin{gathered} \cos O=-\frac{54}{330} \\ \cos O=-0.16363 \end{gathered}$

which gives

$\begin{gathered} \angle O=\cos ^{-1}(-0.16363) \\ \angle O=99.418 \end{gathered}$

Once we have one angle, we can use the law of sines as follows,

$\frac{\sin O}{20}=\frac{\sin D}{15}$

which gives

$\begin{gathered} \frac{\sin 99.418}{20}=\frac{\sin D}{15} \\ \text{then} \\ \sin D=15\times\frac{\sin99.418}{20} \end{gathered}$

so, angle D is

$\begin{gathered} \sin D=0.73989 \\ \angle D=\sin ^{-1}(0.73989) \\ \angle D=47.722 \end{gathered}$

Finally, since the interior angles of any triangle add up to 180, we have

$\begin{gathered} \angle T+\angle D+\angle O=180 \\ \angle T+47.722+99.418=180 \\ \angle T+147.14=180 \\ \angle T=32.86 \end{gathered}$

In summary, the answers are:

$\begin{gathered} \angle D=47.722 \\ \angle O=99.418 \\ \angle T=32.86 \end{gathered}$