Given A=56degrees, C=61degrees, and b=10.5, solve triangle ABC. If no triangle exists, explain why. If two solutions exist, write both. Round your answer to the nearest tenth
This is an "angle-side-angle" specified triangle, and is uniquely defined. The third angle B is obviously 180 - 56 - 61 = 63 degrees.
Use the Law of Sines for the other side lengths. For example:
b/sinB = 11.784 = a/sin A = a/0.8290
a = 9.769, round to 9.8
Similarly for c.
To solve triangle ABC, we will use the law of sines and the law of cosines. The law of sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant. The law of cosines allows us to find the remaining side and angles of a triangle when we know two sides and the included angle.
First, let's check if a triangle can exist with the given information. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's check the conditions for triangle ABC using the given side lengths:
AB + BC > AC
AB + AC > BC
BC + AC > AB
Let's plug in the given values:
AB + BC > AC
AB + AC > BC
BC + AC > AB
10.5 + BC > AC ----(1)
10.5 + AC > BC ----(2)
BC + AC > AB ----(3)
Now, we can solve triangle ABC by finding the missing angle and side length.
Step 1: Finding angle B using the law of sines.
We know that sin(B) / b = sin(C) / c, where b and c are the side opposite angle B and C, respectively.
Using the given values, let's calculate sin(B):
sin(B) / 10.5 = sin(61°) / c
Rearranging the formula, we get sin(B) = (10.5 * sin(61°)) / c
Step 2: Finding angle A using the sum of angles in a triangle.
Since we know angle B and angle C, we can find angle A:
A = 180° - B - C
Step 3: Finding side AC using the law of cosines.
We know that c^2 = a^2 + b^2 - 2ab * cos(C), where a and b are the sides opposite angles A and B, respectively. In this case, we are looking for side AC, which is opposite angle A.
AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(A)
Now, let's calculate and solve for AC using the given values:
AC^2 = (10.5)^2 + BC^2 - 2 * 10.5 * BC * cos(A)
Based on the given information, we can't determine the exact lengths of side BC or angle B. Thus, we need to consider two different scenarios:
Scenario 1: Ambiguous Case (two solutions)
If a triangle exists and one angle is given, there can be two possible triangles when the sine ratio produces two valid values for the missing angle.
In this case, we can use the ambiguous case of the law of sines to find both solutions for side BC:
sin(B) / 10.5 = sin(C) / BC
BC = (10.5 * sin(C)) / sin(B)
Plug in the given values to find the two possible solutions for angle B:
BC1 = (10.5 * sin(61°)) / sin(B)
BC2 = (10.5 * sin(61°)) / sin(B)
Then, use the law of cosines to find the corresponding values of AC for both solutions.
Scenario 2: No triangle exists
If the conditions of the triangle inequality theorem are not satisfied, then no triangle can be formed using the given side lengths.
To determine if a triangle exists, compare each of the three inequalities we set up earlier. If any of these inequalities are not true, then no triangle can exist.
If there is no solution for a triangle, the given side lengths and angles do not form a valid triangle according to the triangle inequality theorem.