In triangle ABC, A=35 degrees, a=43, and c=20. Determine whether ABC has no solution, one solution, or two solutions. Then solve the triangle. Round to the nearest tenth.
no solution.
According to the law of sines,
sin C = c* sin A/a = 0.2668
C = 15.5 degrees, or 164.5 degrees. Only the 15.5 is possible without exceeding 180 degrees for all angles. That means B = 180 - 35 - 15.5 = 129.5 degrees. There is only one solution in this case, although some side-side-angle specified triangles (like this one) have two solutions.
b = sin B* a/sin A = 57.8
I get the work part of it but for this question is it one solution or two?
One
What I tried to say is that sometimes when two adjacent sides of a triangle are specified, along with the angle adjacent to one side, two solutions are possible. This called the "ambiguous Side-Side-Angle" case. This is a Side-Side-Angle case, but it turns out that only one solution is possible anyway. One of the two computed angles from the law of sines is too large to make a triangle possible.
To determine whether triangle ABC has no solutions, one solution, or two solutions, we can 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.
The formula for the Law of Sines is:
a/sin(A) = b/sin(B) = c/sin(C)
In triangle ABC, we know that A = 35 degrees, a = 43, and c = 20. Let's determine whether the equation holds true:
43/sin(35) = b/sin(B) = 20/sin(C)
We are given the lengths of sides a and c, but we are missing one side and angle. To find the measure of angle B, we can use the fact that the sum of the angles in a triangle is 180 degrees:
B = 180 - A - C
B = 180 - 35 - C
Now, let's plug in the values into the equation and evaluate:
43/sin(35) = b/sin(180 - 35 - C) = 20/sin(C)
We can solve for b by cross-multiplying:
43*sin(180 - 35 - C) = 20*sin(35)
However, the sine of an angle can only produce values between -1 and 1. Therefore, there are no possible values for C that would satisfy this equation. Hence, triangle ABC has no solution.
Therefore, there is no triangle that can be formed with the given values.