Why cant we solve an oblique triangle with the Law of Sines if we are given SAS?
sin(x) and sin(180-x) have the same value.
gotta use the law of cosines, which takes into account the change of sign for obtuse angles.
To understand why we can't solve an oblique triangle with the Law of Sines if we are given the Side-Angle-Side (SAS) information, let's first review what the Law of Sines states:
The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, it can be expressed as:
sin(A) / a = sin(B) / b = sin(C) / c
Where A, B, and C are the angles of the triangle, and a, b, and c are the lengths of the respective sides.
When given Side-Angle-Side information, we know the lengths of two sides (let's say a and b) and the measure of an included angle (let's say C). However, to use the Law of Sines, we need to know the ratio between the length of one side and the sine of its opposite angle.
In a right triangle, we can easily determine the remaining side and angles using the given side and angle information, but in an oblique triangle (a triangle that does not have a right angle), the Law of Sines alone is not sufficient to solve the triangle when we are given SAS.
To solve an oblique triangle with SAS information, additional methods such as the Law of Cosines or the Sine Rule (a combination of the Law of Sines and the Law of Cosines) must be used. These methods allows us to find the remaining sides and angles of the triangle by relating the lengths of sides and angles to each other.
In summary, while the Law of Sines is a helpful tool to solve triangles, it is not enough to solve an oblique triangle when given only SAS information. Additional methods like the Law of Cosines or the Sine Rule are required.