consider a triangle $abc$ like the one below. suppose that $c = 123^{\circ}$, $a = 67$, and $b = 4$. (the figure is not drawn to scale.) solve the triangle.

carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth.

if there is more than one solution, use the button labeled \or\.

$a = \box^{\circ}$, $b = \box^{\circ}$, $c = \box$

Question

consider a triangle $abc$ like the one below. suppose that $c = 123^{\circ}$, $a = 67$, and $b = 4$. (the figure is not drawn to scale.) solve the triangle.

carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth.

if there is more than one solution, use the button labeled \or\.

$a = \box^{\circ}$, $b = \box^{\circ}$, $c = \box$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Law of Cosines", "Law of Sines", "Solving Triangles" ], "new_concepts": [], "current_concepts": [ "Law of Cosines", "Law of Sines", "Solving Triangles" ] } ## Apply the Law of Cosines Using the *Law of Cosines* and *Solving Triangles* knowledge points $$ \begin{aligned} c^2 &= a^2 + b^2 - 2ab \cos(C) \ c^2 &= 67^2 + 4^2 - 2(67)(4) \cos(123^\circ) \ c^2 &= 4489 + 16 - 536 \cos(123^\circ) \ c^2 &\approx 4505 - 536 (-0.544639) \ c^2 &\approx 4505 + 291.9265 = 4796.9265 \ c &\approx 69.2600 \approx 69.3 \end{aligned} $$ ## Apply the Law of Sines Using the *Law of Sines* and *Solving Triangles* knowledge points $$ \begin{aligned} \frac{\sin(A)}{a} &= \frac{\sin(C)}{c} \ \sin(A) &= \frac{a \sin(C)}{c} \ \sin(A) &\approx \frac{67 \sin(123^\circ)}{69.2600} \ \sin(A) &\approx \frac{67 (0.838671)}{69.2600} \approx 0.811305 \ A &\approx \arcsin(0.811305) \approx 54.2201^\circ \approx 54.2^\circ \end{aligned} $$ ## Find the remaining angle Using the *Solving Triangles* knowledge point $$ \begin{aligned} B &= 180^\circ - C - A \ B &\approx 180^\circ - 123^\circ - 54.2201^\circ \ B &\approx 2.7799^\circ \approx 2.8^\circ \end{aligned} $$ Consider a triangle $ABC$ like the one below. Suppose that $C = 123^\circ$, $a = 67$, and $b = 4$. Solve the triangle. $A =$ $54.2$$^\circ$, $B =$ $2.8$$^\circ$, $c =$ $69.3$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Solving Triangles" ] }

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QUESTION IMAGE

Question

Explanation:

Apply the Law of Cosines

Apply the Law of Sines

Find the remaining angle

Answer: