Problem
Two sides and an angle are given below. First determine whether the information results in no triangle, one triangle, or two triangles. Solve each resulting triangle. \[ \mathrm{a}=14.7, \mathrm{~b}=15,7 \text {, and } \mathrm{A}=67.6^{\circ} \] Determine the value of $\sin B$. $\sin B=\square$ (Type an integer or decimal rounded to four decimal places as needed.) How many and what types of triangles does the given information produce? no triangle one or two oblique triangles one right triangle Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There Is only one triangle where $\mathrm{B} \approx \square^{\circ}$ (Type an integer or decimal rounded to one decimal place as needed.) B. There are two triangles. The smaller angle is $B_{1} \approx \square^{\circ}$. The larger angle is $B_{2} \approx \square^{\circ}$, (Type integers or decimals rounded to one decimal place as needed.)
Solution
Step 1 ：Use the Law of Sines to find the value of sin B: sin B = (15.7/14.7) * sin 67.6°
Step 2 ：Calculate sin B: sin B ≈ 0.9997
Step 3 ：Use the Law of Sines to find the value of side c: c = (14.7 * sin C) / sin 67.6°
Step 4 ：Determine the range of B values that would result in a valid triangle: B < 112.4°
Step 5 ：Since B = 67.6° is within this range, there is only one triangle that can be formed.
Step 6 ：Therefore, the correct choice is: A. There is only one triangle where B ≈ 67.6°
