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=(15^2+11^2-20^2)/(2(11)(15))`

which gives

`\cos O=(346-400)/(330)`

then

`\begin{gathered} \cos O=-(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,

`(\sin O)/(20)=(\sin D)/(15)`

which gives

`\begin{gathered} (\sin 99.418)/(20)=(\sin D)/(15) \\ \text{then} \\ \sin D=15*(\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}`