insert step 1: Identify the given information in the triangle. We have angle A = 112.8°, side b = 6.28 m, and side c = 12.2 m.
insert step 2: Use the Law of Cosines to find the unknown side a. The formula is: a^2 = b^2 + c^2 - 2bc \cdot \cos(A).
insert step 3: Substitute the known values into the Law of Cosines formula: a^2 = (6.28)^2 + (12.2)^2 - 2 \cdot 6.28 \cdot 12.2 \cdot \cos(112.8°).
insert step 4: Solve for a by calculating the right-hand side of the equation and then taking the square root of the result.
insert step 5: Use the Law of Sines to find the remaining angles B and C. The formula is: \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Law of Sines
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles. This relationship is crucial for solving triangles when given two angles and one side or two sides and a non-included angle. It allows for the calculation of unknown angles and sides, making it a fundamental tool in trigonometry.
The Triangle Sum Theorem asserts that the sum of the interior angles of a triangle is always 180 degrees. This theorem is essential for finding missing angles in a triangle when two angles are known. In the context of the given problem, it can be used to determine the third angle after identifying the known angle.
Solving Right Triangles with the Pythagorean Theorem
Ambiguous Case of the Law of Sines
The ambiguous case of the Law of Sines occurs when two sides and a non-included angle are known, leading to potentially two different triangles, one triangle, or no triangle at all. Understanding this concept is vital when solving triangles, as it helps determine the number of possible solutions and ensures accurate results in ambiguous situations.