Solve the problem.
-Given a triangle with $\mathrm { a } = 9 , \mathrm {~b} = 11 , \alpha = 31 ^ { \circ }$, what is (are) the possible length(s) of $\mathrm { c }$ ? Round your answer to two decimal places.
A) $16.42$ or $2.44$
B) $16.42$ or $3.41$
C) $6.61$
D) $14.21$

Question

Quizplus · Accepted Answer

Using the Law of Sines, we can find the possible lengths of $c$ by setting up the equation $\frac{a}{\sin \alpha} = \frac{c}{\sin \gamma}$. However, since we don't directly know $\gamma$, we first find $\beta$ using the Law of Cosines or another application of the Law of Sines if possible. Given the information, we might not directly find $\beta$ or $\gamma$ without additional steps, but we can calculate $c$ using the Law of Cosines directly: $c^2 = a^2 + b^2 - 2ab\cos(\alpha)$. This might lead to two solutions based on the ambiguous case of the Law of Sines, where two different triangles could be formed (one acute and one obtuse), hence the two possible values for $c$. The correct values, after calculations and considering potential rounding, match choice A.

Solve the Problem a=9, b=11,α=31∘\mathrm { a } = 9 , \mathrm {~b} = 11 , \alpha = 31 ^ { \circ }a=9, b=11,α=31∘

Solve the Problem $\mathrm { a } = 9 , \mathrm {~b} = 11 , \alpha = 31 ^ { \circ }$