In ΔABC, A=75 degrees and b=6. Giving your answer in interval notation, find the range of values of a such that:
only one triangle is possible.
two different triangles are possible.
Please explain or show work!
To determine the range of values for side length "a" in ΔABC such that only one triangle is possible, we can use the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
For only one triangle to be possible, we need to find the range of values for "a" such that the sum of the lengths of sides "b" and "c" is greater than the length of side "a".
Since we know angle A is 75 degrees and side b is 6 units, we can use the Law of Sines to find the range of values for side c. The Law of Sines states:
sin(A) / a = sin(B) / b = sin(C) / c
We can rearrange this equation to solve for side c:
c = (b * sin(C)) / sin(B)
Substituting the given values:
c = (6 * sin(C)) / sin(180 - A - C)
Now we can use the triangle inequality theorem to find the range of values for side a. For only one triangle to be possible, we want:
b + c > a and a + c > b
Substituting the given values and solving for a:
6 + ((6 * sin(C)) / sin(180 - A - C)) > a
a + ((6 * sin(C)) / sin(180 - A - C)) > 6
Simplifying these inequalities will give us the range of values for "a" such that only one triangle is possible.
Similarly, to find the range of values for side "a" such that two different triangles are possible, we need to consider the overlapping range where both the triangle inequality inequalities are met:
6 + ((6 * sin(C)) / sin(180 - A - C)) < a
a + ((6 * sin(C)) / sin(180 - A - C)) < 6
Once we simplify these inequalities, we will have the range of values for side "a" where two different triangles are possible.
It's important to note that the exact ranges will depend on the angle C, as it can vary within certain limits before the triangle inequalities are no longer met.