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

EXPLANATION

We can find B using the law of sines,

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

$\frac{b}{\sin B}=\frac{c}{\sin C}$

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

$\frac{\sin B}{b}=\frac{\sin C}{c}$

Multiply both sides by b,

$\sin B=\frac{b}{c}\sin C$

And take the inverse of the sine,

$B=\sin ^{-1}\mleft(\frac{b}{c}\sin C\mright)$

Replace with the values and solve,

$B=\sin ^{-1}\mleft(\frac{5}{11}\sin 120\degree\mright)\approx23.2\degree$

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,

$\frac{a}{\sin A}=\frac{c}{\sin C}$

Solving for a,

$a=c\cdot\frac{\sin A}{\sin C}=11\cdot\frac{\sin 36.8\degree}{\sin 120\degree}\approx7.6$

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