Explain why we cannot solve an oblique triangle with the Law of Sines given SAS.

Think about an obtuse triangle ABC where we know b and c and A.

Then a/sinA = c/sinC

allows for C to be acute or obtuse

To understand why we cannot solve an oblique triangle with the Law of Sines given SAS (Side-Angle-Side), we need to delve into how the Law of Sines operates and what information is required to apply it effectively.

The Law of Sines relates the ratios of the side lengths of a triangle to the sines of its opposite angles. It states:

a/sin(A) = b/sin(B) = c/sin(C)

where a, b, and c represent the lengths of the triangle's sides, and A, B, and C represent the measures of its angles.

When given SAS, we have two known side lengths and the measure of the included angle. However, this information alone is insufficient to solve the triangle completely.

To understand why, let's consider the possible cases:

1. SSA (Side-Side-Angle): This case may not have a unique solution. Given two side lengths and a non-included angle, there can be two different triangles that satisfy those conditions. Therefore, the measurements of the remaining sides and angles cannot be determined uniquely.

2. SAS (Side-Angle-Side): This case may have a unique solution if the included angle is acute. In this situation, we can determine the remaining angle and then use the Law of Sines to find the remaining side lengths.

However, when the included angle is obtuse, there may be multiple solutions or no solution at all. When applying the Law of Sines, the sine function returns positive values between 0 and 1 for acute angles only. For obtuse angles, the sine function returns a value greater than 1, which does not correspond to a valid length. Therefore, we cannot use the Law of Sines to find the remaining angles or sides accurately in this case.

To solve an oblique triangle with SAS, additional information is required. This could include the measurement of another angle, the measurement of another side length, or another sides-angles combination. With these extra pieces of information, we can apply the Law of Sines or other trigonometric laws to find all the missing angles and side lengths of the triangle.