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
To solve triangle ABC, we can use the Law of Sines or the Law of Cosines depending on the information given.
Given angle A = 56 degrees, angle C = 61 degrees, and side b = 10.5, we can start by finding angle B using the triangle angle sum property:
Angle A + Angle B + Angle C = 180 degrees
56 + Angle B + 61 = 180
Angle B = 180 - 56 - 61
Angle B = 63 degrees
Now we can proceed with the Law of Sines or the Law of Cosines.
1. Law of Sines:
Using the Law of Sines, we have the following formula:
a / sin(A) = b / sin(B) = c / sin(C)
We already know side b and angle A and B, so we can use those to find side a and side c.
a / sin(56) = 10.5 / sin(63)
a / 0.8290 = 10.5 / 0.8910
a = (10.5 * sin(56)) / sin(63)
a ≈ 9.86
c / sin(61) = 10.5 / sin(63)
c / 0.8780 = 10.5 / 0.8910
c = (10.5 * sin(61)) / sin(63)
c ≈ 10.18
So the triangle ABC has sides a ≈ 9.86, b = 10.5, and c ≈ 10.18 with angles A = 56 degrees, B = 63 degrees, and C = 61 degrees.
2. Law of Cosines:
If you prefer to use the Law of Cosines, you can use the formula:
c^2 = a^2 + b^2 - 2ab * cos(C)
Substituting the given values:
c^2 = a^2 + 10.5^2 - 2 * a * 10.5 * cos(61)
Now you can solve for a using this equation with the known values of c, b, and C.
Similarly, you can use the Law of Cosines to find side b or side c, depending on the given information.
If no solution is obtained or if a negative number appears under the square root, then it means that no triangle exists with the given conditions.