Given
b = 0.6013 mm
c = 0.8854 mm
Part A: Solving for angle α
Using the trigonometric function arc cosine, we have the following:
[tex]\begin{gathered} \alpha=\cos^{-1}\Big(\frac{b}{c}\Big) \\ \alpha=\cos^{-1}\Big(\frac{0.6013}{0.8854}\Big) \\ \alpha=47.22445452° \end{gathered}[/tex]Rounding to four significant digits, angle α is 47.22°.
Part B: Solving for angle β
Now that we have angle α, use the fact that the sum of the interior angle of a triangle is equal to 180°.
Angle β therefore is
[tex]\begin{gathered} \alpha+\beta+90°=180° \\ 47.22\degree+\beta+90\degree=180\degree \\ \beta+137.22\degree=180° \\ \beta=180°-137.22\degree \\ \beta=42.78\degree \\ \\ \text{Therefore, }\beta\text{ is equal to }42.78\degree \end{gathered}[/tex]Part C:
Use the Pythagorean theorem to solve for side a, we have the following:
[tex]\begin{gathered} a^2+b^2=c^2 \\ a^2+(0.6013)^2=(0.8854)^2 \\ a^2+0.36156169=0.78393316 \\ a^2=0.78393316-0.36156169 \\ a^2=0.42237147 \\ \sqrt{a^2}=\sqrt{0.42237147} \\ a\approx0.6499011232 \end{gathered}[/tex]Rounding to 4 significant digits, the measurement of a is 0.6499 mm.