Let us start by illustrating the problem using a diagram:
To find the measure of angle A, 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{ = 63.3}^2\text{ + 43.59}^2\text{ - 2 }\times\text{ 63.63 }\times\text{ 43.59 }\times\text{ cos 45.4} \\ c^2\text{ = 2053.837} \\ c\text{ = }\sqrt{2053.837} \\ c\text{ }\approx\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 A
Sine rule is defined as:
[tex]\frac{sin\text{ A}}{a}\text{ = }\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{ A}}{a}\text{ } \\ \frac{sin\text{ 45.4}}{45.32}\text{ = }\frac{sin\text{ A}}{63.63} \\ sin\text{ A = }\frac{sin\text{ 45.4 }\times\text{ 63.63}}{45.32} \\ sin\text{ A = 0.999696} \\ A\text{ }\approx\text{ 88.59} \end{gathered}[/tex]Answer:
Measure of angle A = 88.59 degrees
