Question

question 17 of 20
how many solutions does a triangle with side lengths ( a = 22 ), ( a = 117^{circ} ), and ( b = 25 ) have?
write your answer in numeric form.

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Substituting the given values \(a = 22\), \(A=117^{\circ}\), and \(b = 25\), we get \(\frac{22}{\sin 117^{\circ}}=\frac{25}{\sin B}\).

Step2: Solve for \(\sin B\)

Cross - multiply: \(22\sin B=25\sin 117^{\circ}\). Then \(\sin B=\frac{25\sin 117^{\circ}}{22}\). Calculate \(\sin 117^{\circ}\approx0.891\). So \(\sin B=\frac{25\times0.891}{22}\approx\frac{22.275}{22}\approx1.0125\).

Step3: Analyze the value of \(\sin B\)

Since the range of the sine function is \([- 1,1]\), and \(\sin B\approx1.0125>1\), there is no angle \(B\) that satisfies the equation.

Answer:

\(0\)