Final answer:
The number of solutions for triangle △abc with sides b=27, c=17 and angle C=56° is one. By applying the Law of Sines, we find that there is only one valid angle for B that corresponds to the given dimensions, thus concluding that only one triangle is possible.
Step-by-step explanation:
The student asked about the number of possible solutions for a triangle (△abc) with given lengths of sides b=27 and c=17, and an angle C=56°. To determine the number of possible triangles, we use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. In this case, we first find the sine of angle C and then compare it to the ratio of side b to the sine of its opposite angle B:
Calculate the sine of angle C: {
{ \sin(56°) }b
Compare it to side b: {
{ \frac{27}{\sin(B)} } => \sin(B) = \frac{27\sin(56°)}{17}
We then determine whether this sine value corresponds to a valid angle between 0° and 180° and not equal to 56°, the already known angle C. There will be two solutions if } \sin(B) sequences are less than or equal to 1; there will be one solution if } \sin(B) is exactly equal to 1; and no solution if } \sin(B) is greater than 1.
In this scenario, the calculation will show that there is only one possible solution for triangle △abc, since the value of } \sin(B) calculated will be less than 1 and will correspond to an angle B that is not obtuse (greater than 90°), leaving no room for an additional triangle to exist with the given measurements.