Let us begin by illustrating the problem using a diagram:
To find the measure of angle B, we need to first find the length of the side opposite angle C using cosine rule.
Cosine rule is defined to be:
[tex]c^2\text{ = a}^2\text{ + b}^2\text{ - 2abCosC}[/tex]Substituting the given sides and angle:
Let c be the side opposite angle C, b be the side opposite angle B and a be the side opposite angle A
[tex]\begin{gathered} c^2\text{ = 63.63}^2\text{ + 43.59}^2\text{ - 2}\times\text{ 63.63 }\times\text{ 43.59 }\times\text{ cos45.4} \\ c^2\text{ = 2053.837} \\ c\text{ =}\sqrt{2053.837} \\ c\text{ = 45.32 mi} \end{gathered}[/tex]Hence, we have the triangle:
The next step is to use sine rule to find the measure of angle B
sine rule is defined as:
[tex]\frac{sin\text{ A}}{a}=\frac{sin\text{ B}}{b}\text{ = }\frac{sin\text{ C}}{c}[/tex]Applying sine rule:
[tex]\begin{gathered} \frac{sin\text{ C}}{c}=\text{ }\frac{sin\text{ B}}{b} \\ \frac{sin\text{ 45,4}}{45.32\text{ }}\text{ = }\frac{sin\text{ B}}{43.59} \\ sin\text{ B= }\frac{sin45.4\text{ }\times\text{ 43.49}}{45.32} \\ sin\text{ B = 0.68327} \\ B=\text{ }\sin^{-1}0.68327 \\ B\text{ = 43.0997} \\ B\text{ }\approx\text{ 43.1}^0 \end{gathered}[/tex]Answer:
Measure of angle B = 43.1 degrees
