Solve the triangle. Find the angles  and  first.
-$$\mathrm { a } = 8 , \mathrm {~b} = 6 , \mathrm { c } = 4$$
A) $\alpha = 104.5 ^ { \circ } , \beta = 46.6 ^ { \circ } , \gamma = 28.9 ^ { \circ }$
B) $\alpha = 104.5 ^ { \circ } , \beta = 28.9 ^ { \circ } , \gamma = 46.6 ^ { \circ }$
C) $\alpha = 46.6 ^ { \circ } , \beta = 28.9 ^ { \circ } , \gamma = 104.5 ^ { \circ }$
D) $\alpha = 46.6 ^ { \circ } , \beta = 104.5 ^ { \circ } , \gamma = 28.9 ^ { \circ }$

Question

Quizplus · Accepted Answer

Using the Law of Cosines, we can find the angles of the triangle:$$\cos \alpha = \frac{b^2 + c^2 - a^2}{2bc}$$$$\cos \alpha = \frac{6^2 + 4^2 - 8^2}{2(6)(4)}$$$$\cos \alpha = \frac{36 + 16 - 64}{48}$$$$\cos \alpha = \frac{-12}{48}$$$$\alpha = \cos^{-1}\left(\frac{-12}{48}\right)$$$$\alpha = 104.5^\circ$$Using the Law of Sines, we can find the other angles of the triangle:$$\frac{\sin \beta}{b} = \frac{\sin \alpha}{a}$$$$\frac{\sin \beta}{6} = \frac{\sin 104.5^\circ}{8}$$$$\sin \beta = \frac{6 \sin 104.5^\circ}{8}$$$$\beta = \sin^{-1}\left(\frac{6 \sin 104.5^\circ}{8}\right)$$$$\beta = 46.6^\circ$$$$\gamma = 180^\circ - \alpha - \beta$$$$\gamma = 180^\circ - 104.5^\circ - 46.6^\circ$$$$\gamma = 28.9^\circ$$Therefore, the correct answer is A: $\alpha = 104.5 ^ { \circ } , \beta = 46.6 ^ { \circ } , \gamma = 28.9 ^ { \circ }$.

Solve the Triangle a=8, b=6,c=4\mathrm { a } = 8 , \mathrm {~b} = 6 , \mathrm { c } = 4a=8, b=6,c=4

Solve the Triangle $\mathrm { a } = 8 , \mathrm {~b} = 6 , \mathrm { c } = 4$