Question

using the law of sines for the ambiguous case
△jkl has ( j = 7 ), ( k = 11 ), and ( mangle j = 18^circ ). complete the statements to determine all possible measures of angle ( k ).
triangle jkl meets the ssa 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 with options: ( 7sin(k) ), ( 11sin(k) ), ( ksin(7) )
solve for the measure of angle... calculator to determine the value.
round to the nearest degree...
however, because this is the ambiguous case, the measure of angle ( k ) could also be dropdown.

Explanation:

Step1: Recall Law of Sines Cross - Multiply

The Law of Sines is given by $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$ for a triangle with angles $A,B,C$ and opposite sides $a,b,c$ respectively. In our case, we have $\frac{\sin(18^{\circ})}{7}=\frac{\sin(K)}{11}$. When we cross - multiply (multiply both sides by $7\times11$), we get $11\times\sin(18^{\circ}) = 7\times\sin(K)$. So the expression in the box should be $7\sin(K)$.

Step2: Solve for $\sin(K)$

From $11\sin(18^{\circ})=7\sin(K)$, we can solve for $\sin(K)$ by dividing both sides by 7: $\sin(K)=\frac{11\sin(18^{\circ})}{7}$.
First, calculate $\sin(18^{\circ})\approx0.3090$. Then $11\times0.3090 = 3.399$. Then $\frac{3.399}{7}\approx0.4856$.

Step3: Find the first angle $K$

Now, we find the angle whose sine is approximately $0.4856$. Using the inverse sine function, $K=\sin^{-1}(0.4856)\approx29^{\circ}$ (rounded to the nearest degree).

Step4: Find the second angle (Ambiguous Case)

In the SSA ambiguous case, if $\sin(K)=s$ and $K$ is an acute angle, then another possible angle is $180^{\circ}-K$. 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 $K = 151^{\circ}$ and $J=18^{\circ}$, then the third angle $L=180^{\circ}-151^{\circ}-18^{\circ}=11^{\circ}$, which is positive, so it is a valid angle.

Answer:

• The value in the cross - multiply box is $7\sin(K)$.
• The first measure of angle $K$ (rounded to nearest degree) is $29^{\circ}$.
• The other possible measure of angle $K$ is $151^{\circ}$.