Question

Round your answers to four significant digits. Write your answers for angle measures in decimal degrees.

Answer 1

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:

`\begin{gathered} \alpha=\cos^(-1)\Big((b)/(c)\Big) \\ \alpha=\cos^(-1)\Big((0.6013)/(0.8854)\Big) \\ \alpha=47.22445452° \end{gathered}`

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

`\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}`

Part C:

Use the Pythagorean theorem to solve for side a, we have the following:

`\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 \\ √(a^2)=√(0.42237147) \\ a\approx0.6499011232 \end{gathered}`

Rounding to 4 significant digits, the measurement of a is 0.6499 mm.