Question

Find all the missing elements:C = 120°b = 5C = 11A = [?]° B = [ ]º a = [

Answer 1

ANSWERS

• A = 36.8°

,

• B = 23.2°

,

• a = 7.6

Step-by-step explanation

We can find B using the law of sines,

We have b = 5, c = 11 and C = 120°. Using the last two ratios,

`(b)/(\sin B)=(c)/(\sin C)`

Rise both sides of the equation to -1 - i.e. flip both sides,

`(\sin B)/(b)=(\sin C)/(c)`

Multiply both sides by b,

`\sin B=(b)/(c)\sin C`

And take the inverse of the sine,

`B=\sin ^(-1)\mleft((b)/(c)\sin C\mright)`

Replace with the values and solve,

`B=\sin ^(-1)\mleft((5)/(11)\sin 120\degree\mright)\approx23.2\degree`

Then, knowing that the sum of the measures of all the interior angles of a triangle is 180°, we find A,

`A+B+C=180\degree`

Solving for A,

`A=180\degree-B-C=180\degree-23.2\degree-120\degree\approx36.8\degree`

Finally, let's find a using the law of sines with the first and last ratios,

`(a)/(\sin A)=(c)/(\sin C)`

Solving for a,

`a=c\cdot(\sin A)/(\sin C)=11\cdot(\sin 36.8\degree)/(\sin 120\degree)\approx7.6`

Hence, the missing elements are A = 36.8°, B = 23.2° and a = 7.6.