g) given triangle $\delta mnp$ where $\angle m = 25^\circ$, $n = 10$, and $m = 7$. solve for $\angle p$ (nearest tenth)

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Law of Sines", "Ambiguous Case of Law of Sines", "Triangle Angle Sum" ], "current_concepts": [ "Law of Sines", "Ambiguous Case of Law of Sines", "Triangle Angle Sum" ] } ## Apply the Law of Sines to find angle N $$ \begin{aligned} &\frac{\sin(N)}{n} = \frac{\sin(M)}{m} \ &\frac{\sin(N)}{10} = \frac{\sin(25^\circ)}{7} \ &\sin(N) = \frac{10 \cdot \sin(25^\circ)}{7} \approx \frac{10 \cdot 0.422618}{7} \approx 0.60374 \ &N_1 = \arcsin(0.60374) \approx 37.1^\circ \ &N_2 = 180^\circ - 37.1^\circ = 142.9^\circ \end{aligned} $$ ## Check validity of both cases $$ \begin{aligned} &\text{Case 1: } N_1 = 37.1^\circ \ &M + N_1 = 25^\circ + 37.1^\circ = 62.1^\circ < 180^\circ \quad (\text{Valid}) \ &\text{Case 2: } N_2 = 142.9^\circ \ &M + N_2 = 25^\circ + 142.9^\circ = 167.9^\circ < 180^\circ \quad (\text{Valid}) \end{aligned} $$ ## Calculate the corresponding values for angle P $$ \begin{aligned} &P_1 = 180^\circ - (M + N_1) = 180^\circ - 62.1^\circ = 117.9^\circ \ &P_2 = 180^\circ - (M + N_2) = 180^\circ - 167.9^\circ = 12.1^\circ \end{aligned} $$ $117.9^\circ$ or $12.1^\circ$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Law of Sines" ] }

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Explanation:

Apply the Law of Sines to find angle N

Check validity of both cases

Calculate the corresponding values for angle P

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