Answer:
A.) one solution; c ≈ 4,1; B ≈ 29,7°; C = 30,3°
Step-by-step explanation:
We will be using the Law of Sines:
Solving for Angle Measures
[tex]\frac{sin∠C}{c} = \frac{sin∠B}{b} = \frac{sin∠A}{a}[/tex]
In the end, use the [tex]sin^{-1}[/tex] function or else you will throw off your answer.
Solving for Sides
[tex]\frac{c}{sin∠C} = \frac{b}{sin∠B} = \frac{a}{sin∠A}[/tex]
Given instructions:
120° = A
7 = a
4 = b
Now, we have to solve for m∠B, since its side has already been defined, also, it is because angle A and side a have all information filled in:
[tex]\frac{sin\: ∠B}{4} = \frac{sin\: 120°}{7} \\ \\ \frac{4sin\: 120°}{7} = sin\: ∠B \\ \\ \frac{2\sqrt{3}}{7} = sin\: ∠B \\ \\ *\: sin^{-1} \frac{2\sqrt{3}}{7} ≈ 29,66128776° \\ \\ 29,7° ≈ m∠B[/tex]
Now that we have the measure of the second angle, we can use the Triangular Interior Angles Theorem to find the third angle measure:
[tex]29,7° + 120° + C = 180°[/tex]
149,7° + [tex]C[/tex] = 180°
-149,7° - 149,7°
_____________________________
[tex]30,3° = C[/tex]
Now, we have to find side c. We could use the information for angle B and side b:
[tex]\frac{c}{30,3} = \frac{4}{29,7} \\ \\ \frac{4 \times 30,3}{29,7} = \frac{121,2}{29,7} = 4\frac{8}{99}\\ \\ 4,1 ≈ c[/tex]
Now, everything has been defined!
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