Question

what is the measure of ∠c to the nearest whole degree?
70°
77°
80°
85°
law of cosines: a² = b² + c² - 2bccos(a)

Explanation:

Step1: Identify the sides for cosine - rule application

In right - triangle ABC, we know \(a = 8.77\), \(b = 2\), \(c = 9\). We want to find \(\angle C\). Using the law of cosines \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\).

Step2: Rearrange the law of cosines formula to solve for \(\cos(C)\)

\(\cos(C)=\frac{a^{2}+b^{2}-c^{2}}{2ab}\). Substitute \(a = 8.77\), \(b = 2\), and \(c = 9\) into the formula.
\[

$$\begin{align*} a^{2}&=(8.77)^{2}=76.9129\\ b^{2}&=2^{2} = 4\\ c^{2}&=9^{2}=81\\ 2ab&=2\times8.77\times2 = 35.08 \end{align*}$$

\]
\[

$$\begin{align*} \cos(C)&=\frac{76.9129 + 4-81}{35.08}\\ &=\frac{80.9129 - 81}{35.08}\\ &=\frac{- 0.0871}{35.08}\approx - 0.00248 \end{align*}$$

\]

Step3: Find the angle \(C\)

\(C=\cos^{-1}(-0.00248)\approx90^{\circ}\). But we can also use the right - triangle trigonometry since \(\triangle ABC\) is a right - triangle. \(\sin(C)=\frac{AB}{AC}\). Here \(AB = 8.77\) and \(AC = 9\).
\(\sin(C)=\frac{8.77}{9}\approx0.9744\)
\(C=\sin^{-1}(0.9744)\approx77^{\circ}\)

Answer:

\(77^{\circ}\)