Question

using the law of sines for the ambiguous case
△jkl has j = 7, k = 11, and m∠j = 16°. 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: $\frac{sin(16^{circ})}{7}=\frac{sin(k)}{11}$
cross multiply: 11sin(16°) =
solve for the measure of angle k, and use a calculator to determine the value.
round to the nearest degree: m∠k ≈
however, because this is the ambiguous case, the measure of angle k could also be

Explanation:

Step1: Apply law of sines cross - multiplication

Given $\frac{\sin(16^{\circ})}{7}=\frac{\sin(K)}{11}$, cross - multiplying gives $11\sin(16^{\circ}) = 7\sin(K)$.

Step2: Solve for $\sin(K)$

$\sin(K)=\frac{11\sin(16^{\circ})}{7}$.
Using a calculator, $\sin(K)=\frac{11\times0.275637}{7}=\frac{3.032007}{7}\approx0.433144$.

Step3: Find the first value of $K$

$K_1=\sin^{- 1}(0.433144)\approx26^{\circ}$.

Step4: Find the second value of $K$ (ambiguous case)

Since $\sin\theta=\sin(180^{\circ}-\theta)$ in the range of $0^{\circ}<\theta < 180^{\circ}$, the second possible value of $K$ is $K_2 = 180^{\circ}-26^{\circ}=154^{\circ}$.

Answer:

$11\sin(16^{\circ}) = 7\sin(K)$; $m\angle K\approx26^{\circ}$; $m\angle K$ could also be $154^{\circ}$