Final answer:
Using the Law of Sines, angle C in the triangle can be calculated. Given the side lengths and one angle, the sine rule leads to a solution of C ≈ 55.2 degrees for the obtuse angle option. Correct option is B) C ≈ 55.2 degrees
Step-by-step explanation:
To find the angle C in the first triangle where A = 35 degrees, a = 12, and c = 18, acknowledging the ambiguous case, we'll need to use the Law of Sines.
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant for all three sides. This can be written as:
a/sin(A) = c/sin(C) => 12/sin(35 degrees) = 18/sin(C)
First, we calculate sin(35 degrees), and then manipulate the equation to solve for sin(C):
sin(C) = (18 * sin(35 degrees))/12
To find C, take the inverse sine (sin-1) of sin(C). However, because of the ambiguous case, there could be two different angles for C that satisfy the Law of Sines. We need to check for both angles (using the sine curve or the properties of the unit circle) that are in range (0, 180).
Considering the given choices, angle C could either be an acute angle or an obtuse angle. Since the sum of angles in a triangle must equal 180 degrees, and we already have one angle of 35 degrees, we subtract this from 180 degrees to find the possible range for the obtuse angle.
180 degrees - 35 degrees - 90 degrees (minimum angle for obtuse) gives us 55 degrees as the smallest possible obtuse angle C could be. Therefore, we can rule out options C and D, as 31.2 degrees would be acute and 63.8 degrees would not be the smallest possible obtuse angle. Using a calculator, C ≈ 55.2 degrees is the correct obtuse angle, aligning with option B.