Questions: A triangular brace has an angle measure of 92 degrees, with a side opposite this angle measuring 10 inches. The base of the triangular brace, which is adjacent to the given angle measure, is 12 inches in length. Which of the following statements is correct? A. There is not a solution for the angle opposite the side measuring 12 inches. B. The angle opposite the side measuring 12 inches has one solution of approximately 24 degrees. C. The angle opposite the side measuring 12 inches has two solutions of approximately 24 degrees and 36 degrees D. The angle opposite the side measuring 12 inches has one solution of approximately 32 degrees.

Solution Steps

We are given a triangle with an angle \( A = 92^\circ \) and the lengths of the sides opposite this angle \( a = 10 \) inches and adjacent to it \( b = 12 \) inches. We need to find the angle \( B \) opposite the side measuring 12 inches.

Using the Law of Sines, we have the relationship: \[ \frac{\sin(A)}{a} = \frac{\sin(B)}{b} \] Substituting the known values: \[ \frac{\sin(92^\circ)}{10} = \frac{\sin(B)}{12} \]

Rearranging the equation gives: \[ \sin(B) = \frac{\sin(92^\circ) \cdot 12}{10} \] Calculating \( \sin(92^\circ) \) yields a value greater than 1 when multiplied by \( \frac{12}{10} \), leading to: \[ \sin(B) \approx 1.1993 \]

Since \( \sin(B) \) must be in the range \([-1, 1]\), the calculated value \( 1.1993 \) indicates that there is no valid angle \( B \) that satisfies this condition. Therefore, there is not a solution for the angle opposite the side measuring 12 inches.

Final Answer