Question

question 20 of 26
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 32 degrees.

c. the angle opposite the side measuring 12 inches has one solution of approximately 24 degrees.

d. the angle opposite the side measuring 12 inches has two solutions of approximately 24 degrees and 36 degrees

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that for a triangle with sides \(a\), \(b\), \(c\) opposite angles \(A\), \(B\), \(C\) respectively, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let the angle of \(92^\circ\) be \(A\), the side opposite to it (\(a\)) be \(10\) inches, the side of \(12\) inches be \(b\), and the angle opposite to \(b\) be \(B\). So we have \(\frac{a}{\sin A}=\frac{b}{\sin B}\).

Step2: Substitute the known values

Substitute \(a = 10\), \(A=92^\circ\), \(b = 12\) into the Law of Sines formula: \(\frac{10}{\sin(92^\circ)}=\frac{12}{\sin B}\).

Step3: Solve for \(\sin B\)

First, calculate \(\sin(92^\circ)\approx\sin(90^\circ + 2^\circ)=\cos(2^\circ)\approx0.9994\). Then, from \(\frac{10}{0.9994}=\frac{12}{\sin B}\), we can cross - multiply to get \(10\sin B=12\times0.9994\). So \(\sin B=\frac{12\times0.9994}{10}=\frac{11.9928}{10} = 1.19928\).

Step4: Analyze the value of \(\sin B\)

We know that the range of the sine function is \([- 1,1]\). Since \(\sin B=1.19928>1\), there is no real - valued angle \(B\) that satisfies this equation. So there is no solution for the angle opposite the side measuring \(12\) inches.

Answer:

A. There is not a solution for the angle opposite the side measuring 12 inches.