Answer:
To solve for all possible triangles using the given angle A, side a, and side b, apply the Law of Sines to determine the sine of angle B. Then, check the range for valid values, solve for angle B for both acute and obtuse cases, calculate angle C, and find side c using the appropriate law.
Step-by-step explanation:
To solve for all possible triangles given ∠A = 117°, a = 32, and b = 11 using the Law of Sines, one must determine whether a triangle can exist with these measurements. First, consider the formula from the Law of Sines: a/sin(A) = b/sin(B).
Substitute the given values into the formula to find sin(B):
32/sin(117°) = 11/sin(B)
It simplifies to: sin(B) = (11*sin(117°))/32
Calculate sin(B) and check if it's within the range of -1 to 1, since that's the range of sin values. If sin(B) is outside this range, then no such triangle exists and the answer is DNE.
If a valid value of sin(B) is obtained, there may be two possible solutions for B because sin is positive in both the first and second quadrants: one acute angle and one obtuse angle (keeping in mind that the sum of angles in a triangle is 180°).
For each valid B, use 180° - A - B to determine ∠C. The side c can then be found using either the Law of Sines or the Law of Cosines, depending on the information available after finding angles B and C.