Question

using the law of sines for the ambiguous case
δjkl has j = 7, k = 11, and m∠j = 18°. complete the statements to determine all possible measures of angle k.
triangle jkl meets the ▼ 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: (11sin(18^circ) = ▼).
solve for the measure of angle k, and use a calculator to determine the value.
round to the nearest degree: (mangle k = ▼^circ).
however, because this is the ambiguous case, the measure of angle k could also be ▼.

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, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In \(\triangle JKL\), we have \(j = 7\) (opposite \(\angle J\)), \(k = 11\) (opposite \(\angle K\)), and \(m\angle J=18^\circ\). So, \(\frac{j}{\sin J}=\frac{k}{\sin K}\), which gives \(\frac{7}{\sin 18^\circ}=\frac{11}{\sin K}\).

Step2: Cross - multiply to solve for \(\sin K\)

Cross - multiplying the equation \(\frac{7}{\sin 18^\circ}=\frac{11}{\sin K}\) gives \(7\sin K=11\sin 18^\circ\). Then, \(\sin K=\frac{11\sin 18^\circ}{7}\).

Step3: Calculate the value of \(\sin 18^\circ\)

We know that \(\sin 18^\circ\approx0.3090\). Substitute this value into the formula for \(\sin K\): \(\sin K=\frac{11\times0.3090}{7}=\frac{3.399}{7}\approx0.4856\).

Step4: Find the measure of \(\angle K\)

To find \(\angle K\), we take the inverse sine (arcsin) of \(0.4856\). So, \(m\angle K=\arcsin(0.4856)\approx29^\circ\) (rounded to the nearest degree).

Step5: Consider the ambiguous case

In the ambiguous case (SSA), if \(\sin\theta = x\), then another possible angle is \(180^\circ-\theta\). So, the other possible measure of \(\angle K\) 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 \(\angle K = 151^\circ\) and \(\angle J=18^\circ\), then \(\angle L=180^\circ-(151^\circ + 18^\circ)=11^\circ\), which is positive, so this is a valid angle.

Answer:

First, the measure of \(\angle K\) (rounded to the nearest degree) is \(29^\circ\), and the other possible measure (due to the ambiguous case) is \(151^\circ\). For the cross - multiplication step: \(11\sin(18^\circ)\approx3.399\) (since \(11\times\sin(18^\circ)=11\times0.3090 = 3.399\)). The measure of \(\angle K\) (first value) is \(29^\circ\), and the other possible measure is \(151^\circ\).