Everyone, in this example, we're going to tackle a problem where we're given a triangle with side a equals 1, side b equals 4, and angle A equals 30 degrees. If you haven't noticed, we are provided with two sides, a and b, and the angle A, which corresponds with one of those sides. This scenario is a classic example of an SSA (Side-Side-Angle) triangle. If ever in doubt, it's helpful to sketch a quick triangle, labeling side a and side b, with angle A opposite its corresponding side, confirming the SSA relationship. This setup frequently leads to the ambiguous case, where the exact nature of the triangle is unclear, but a sketch can help confirm your analysis.
Let’s proceed to solve this type of triangle without initially drawing it. The first step involves setting up the Law of Sines to determine a second angle. This can be expressed as: A a = B b We only focus on angles A and B and sides a and b due to the absence of data about angle C or side c. By rearranging and substituting the known values, the equation simplifies to: B = b * A a Substituting the numbers results in: B = 4 * 30° 1 = 2
This outcome reveals that the sine of angle B equals 2, which is impossible since sine values range from -1 to 1. Therefore, this problem does not have a solution. To understand why, consider the triangle setup: angle A is 30 degrees, side b (opposite angle A) is 4, and side a is 1. Side a is too short to complete the triangle irrespective of its orientation, which confirms why there is no solution. Hence, no further calculations are needed.