Question

using the law of sines for the ambiguous case
△jkl has j=7, k=11, and m∠j=18°. complete the statements to determine all possible measures of angle k.
triangle jkl meets the dropdown criteria, which means it is the ambiguous case.
substitute the known values into the law of sines: $\frac{sin(18^circ)}{7} = \frac{sin(k)}{11}$.
cross multiply: $11 sin(18^circ)$ dropdown.
solve for the measure of angle k, and use a calculator to determine the value.
round to the nearest degree: $mangle k \approx$ dropdown $^circ$.
however, because this is the ambiguous case, the measure of angle k could also be dropdown $^circ$.

Explanation:

Step1: Identify the case

The triangle has two sides (\(j = 7\), \(k = 11\)) and a non - included angle (\(m\angle J=18^{\circ}\)), so it is the SSA (Side - Side - Angle) case.

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\) for a triangle with angles \(A\), \(B\), \(C\) and opposite sides \(a\), \(b\), \(c\) respectively. For \(\triangle JKL\), we have \(\frac{\sin(18^{\circ})}{7}=\frac{\sin(K)}{11}\).

Step3: Cross - multiply and solve for \(\sin(K)\)

Cross - multiplying gives \(11\sin(18^{\circ}) = 7\sin(K)\). Then \(\sin(K)=\frac{11\sin(18^{\circ})}{7}\).
We know that \(\sin(18^{\circ})\approx0.3090\), so \(\sin(K)=\frac{11\times0.3090}{7}=\frac{3.399}{7}\approx0.4856\).

Step4: Find the first angle \(K\)

Using the inverse sine function, \(K=\sin^{- 1}(0.4856)\approx29^{\circ}\) (rounded to the nearest degree).

Step5: Find the second possible angle \(K\)

In the SSA ambiguous case, if \(\sin(K) = s\), then another possible angle is \(180^{\circ}-\sin^{-1}(s)\). So the other possible angle is \(180^{\circ}- 29^{\circ}=151^{\circ}\). We need to check if this angle is valid. The sum of angles in a triangle is \(180^{\circ}\). If \(K = 151^{\circ}\) and \(J=18^{\circ}\), then the third angle \(L=180^{\circ}-151^{\circ}-18^{\circ}=11^{\circ}\), which is positive, so this angle is valid.

Answer:

• The triangle meets the SSA criteria.
• \(\sin(K)=\frac{11\sin(18^{\circ})}{7}\approx0.4856\)
• \(m\angle K\approx29^{\circ}\)
• The other possible measure of angle \(K\) is \(151^{\circ}\)