Question

Find the side and measure B

Answer 1

When angle opposite to the unknown sides and other two sides are given then we use law of cosines

Law of cosine

$c^2 = a^2 + b^2 - 2ab cos(c)$

From the given diagram

$AB^2 = CB^2 + AC^2 - 2(CB)(AC) cos( c)$

CB = 108

AC= 55

Angle c= 59

$AB^2 = 108^2 + 55^2 - 2(108)(55) cos( 59 )$

$AB^2 = 8570.34767$

Take square root on both sides

AB = 92.6 m

To find out angle B we use sine law

$\frac{sin(a)}{a} = \frac{sin b}{b} = \frac{sin c}{c}$

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

From the figure

$\frac{sin B}{AC} = \frac{sin C}{AB}$

$\frac{sin B}{55} = \frac{sin 59}{92.6}$

sin(B) = 0.50916647

B = $sin^{-1}$(0.50916647)

Angle B= 30.61 degrees