How many solutions does a triangle with values a = 42, A = 117 degrees, and b = 34 have
There is actually one solution
let's use the sine law ...
sin B/34 = sin 117°/42
sin B = 34sin117/42 = .72129...
B = 46.2° or by the CAST rule B could be 133.8°
But, that would mean the angle has 2 obtuse angle, which is not possible
so angle B = 46.2° and there is only one triangle possible
One set of solutions.
To determine how many solutions a triangle has with the given values, we can use the Law of Sines and the fact that the sum of the angles in a triangle is always 180 degrees.
The Law of Sines states that for any triangle with sides a, b, and c, and corresponding angles A, B, and C, the following ratio holds true:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we know that a = 42, A = 117 degrees, and b = 34.
Let's use the Law of Sines to find side c:
42/sin(117 degrees) = 34/sin(B)
To solve for sin(B), we can rearrange the equation:
sin(B) = (34 * sin(117 degrees)) / 42
Now, we can use the inverse sine function to find the measure of angle B:
B = arcsin((34 * sin(117 degrees)) / 42)
Calculating this using a scientific calculator or an online tool, we find that angle B is approximately 31.96 degrees.
To find angle C, we can subtract angles A and B from 180 degrees:
C = 180 degrees - A - B
C = 180 degrees - 117 degrees - 31.96 degrees
C = 30.04 degrees
Now, we have determined all three angles of the triangle: A = 117 degrees, B = 31.96 degrees, and C = 30.04 degrees.
According to the Triangle Sum Theorem, the sum of the angles in a triangle is always 180 degrees. Let's verify if this holds true for our triangle:
A + B + C = 117 degrees + 31.96 degrees + 30.04 degrees
A + B + C = 179.96 degrees
Given that our calculations might have some rounding error, we can conclude that the sum is very close to 180 degrees. Therefore, we have found ONE valid solution for the triangle.
In summary, there is one solution for a triangle with values a = 42, A = 117 degrees, and b = 34.