see the attached figure to better understand the problem
Step 1
Find the value of c
we know that
Applying the Pythagorean Theorem
[tex]c^{2} =a^{2}+b^{2}[/tex]
we have
[tex]a=505\ mm[/tex]
[tex]b=286\ mm[/tex]
substitute the values in the formula
[tex]c^{2} =505^{2}+286^{2}[/tex]
[tex]c^{2} =336,821[/tex]
[tex]c=580.36\ mm[/tex]
Step 2
Find the value of angle A (α)
we know that
in the right triangle ABC
[tex]cos(A)=\frac{b}{c} \\ \\cos(A)=(286/580.36)\\ \\A=arc\ cos(286/580.36)\\ \\A=60.48\ degrees[/tex]
Step 3
Find the angle B (β)
we know that
in the right triangle ABC
angle A and angle B are complementary angles
so
A+B=90
solve for B
B=90-A-------> B=90-60.48-----> B=29.52°
therefore
the answers are
a) the measure of side c is 580.36 mm
b) the measure of the angle α is 60.48°
c) the measure of the angle β is 29.52°
