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Chapter 8: Problem 16

Solve the triangle, if possible. $$B=115^{\circ}, c=45.6 \mathrm{yd}, b=23.8 \mathrm{yd}$$

Short Answer

Expert verified
The triangle is not possible because \( \sin(C) \) exceeds 1.

Step by step solution

01

Identify Known Values

Identify the given values in the triangle: Angle: \( B = 115^{\text{掳}} \) Side: \( c = 45.6 \text{ yd} \) Side: \( b = 23.8 \text{ yd} \)
02

Use the Law of Sines

Use the Law of Sines to find another angle. The Law of Sines states: \[ \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
03

Solve for \( \sin(C) \)

Rearrange the formula to isolate \( \sin(C) \): \[ \sin(C) = \frac{c \sin(B)}{b} \] Substitute the given values: \[ \sin(C) = \frac{45.6 \sin(115^{\text{掳}})}{23.8} \]Calculate: \[ \sin(115^{\text{掳}}) \approx 0.9063 \] \[ \sin(C) = \frac{45.6 \cdot 0.9063}{23.8} \approx 1.736 \] Since \( \sin(C) \) cannot be greater than 1, there is an error, implying that the triangle is not possible.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle-Side Relationships
Understanding angle-side relationships .Are calculated the length of the sides. erived

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