Question

determine the number of triangles abc possible with the given parts. a = 39.4° a = 3.4 c = 8.7 how many possible solutions does this triangle have?

Explanation:

Step1: Apply the Law of Sines

By the Law of Sines, $\frac{a}{\sin A}=\frac{c}{\sin C}$. Substitute $A = 39.4^{\circ}$, $a = 3.4$, and $c = 8.7$:
$$\sin C=\frac{c\sin A}{a}=\frac{8.7\sin39.4^{\circ}}{3.4}$$
Calculate $\sin39.4^{\circ}\approx0.6356$. Then $\sin C=\frac{8.7\times0.6356}{3.4}\approx\frac{5.52}{3.4}\approx1.62$.

Step2: Analyze the value of $\sin C$

Since the range of the sine function is $[- 1,1]$, and $\sin C\approx1.62>1$.

Answer:

$0$