In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. a = 6, c = 5, B = 50°

검증된 단계별 안내

insert step 1: Identify the given elements of the triangle. We have side a = 6, side c = 5, and angle B = 50°.

insert step 2: Use the Law of Cosines to find side b. The formula is b^2 = a^2 + c^2 - 2ac \(\cdot\) \(\cos\)(B).

insert step 3: Substitute the known values into the Law of Cosines formula: b^2 = 6^2 + 5^2 - 2 \(\cdot\) 6 \(\cdot\) 5 \(\cdot\) \(\cos\)(50°).

insert step 4: Solve for b by calculating the right side of the equation and then taking the square root of the result.

insert step 5: Use the Law of Sines to find another angle, such as angle A. The formula is \(\frac{a}{\sin(A)}\) = \(\frac{b}{\sin(B)}\).

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주요 개념

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Law of Sines

The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of its angles. It states that for any triangle, the ratio of a side length to the sine of its opposite angle is constant. This law is particularly useful for solving triangles when given two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA).

Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. This theorem is essential for finding unknown angles in a triangle when two angles are known. In the context of the given problem, it allows us to calculate the third angle once we have determined the other two angles.

Side-Angle Relationships

In any triangle, the lengths of the sides are directly related to the measures of the angles opposite those sides. Specifically, the larger the angle, the longer the side opposite it. This relationship is crucial when using the Law of Sines or the Law of Cosines to solve for unknown sides or angles, as it helps to establish the correct relationships between the given values.