Question

Use the Law of Sines and/ or the Law of Cosines to solve each triangle round to the nearest tenths when necessary

measure of angle P

measure of angle Q

Measure of angle R

Answer 1

Part a) The measure of angle P is
`29.9\°`

Part b) The measure of angle R is
`26.3\°`

Part c) The measure of angle Q is
`123.8\°`

Explanation:

step 1

Find the measure of angle P

Applying the law of cosines

`cos(P)=[q^(2)+r^(2)-p^(2)]/[2qr]`

we have

`p=27\ mi`

`q=45\ mi`

`r=24\ mi`

substitute

`cos(P)=[45^(2)+24^(2)-27^(2)]/[2(45)(24)]=0.8667`

`P=arccos(0.8667)=29.9\°`

Step 2

Find the measure of angle R

Applying the law of sines

`(p)/(sin(P)) =(r)/(sin(R))`

substitute the values and solve for sin(R)

we have

`p=27\ mi`

`r=24\ mi`

`P=29.9\°`

substitute

`(27)/(sin(29.9\°)) =(24)/(sin(R))`

`sin(R)=sin(29.9\°)*(24)/27=0.4431`

`R=arcsin(0.4431)=26.3\°`

step 3

Find the measure of angle Q

we know that

The sum of the interior angles in a triangle must be equal to 180 degrees

so

`m<P+m<Q+m<R=180\°`

substitute the values

`29.9\°+26.3\°+m<Q=180\°`

`m<Q=180\°-(29.9\°+26.3\°)=123.8\°`