In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.
B = 125°, a = 8 yards, c = 5 yards

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Identify the given elements of the triangle: angle \(B = 125^\circ\), side \(a = 8\) yards (opposite angle \(A\)), and side \(c = 5\) yards (opposite angle \(C\)).

Use the Law of Cosines to find the length of side \(b\) (opposite angle \(B\)) since you know two sides and the included angle \(B\). The Law of Cosines formula is: \[b^2 = a^2 + c^2 - 2 \times a \times c \times \cos(B)\]

Calculate \(b\) by taking the square root of the result from the Law of Cosines: \[b = \sqrt{a^2 + c^2 - 2ac \cos(B)}\]

Use the Law of Sines to find one of the other angles, for example angle \(A\), using the formula: \[\frac{\sin(A)}{a} = \frac{\sin(B)}{b}\] Rearranged to solve for \(\sin(A)\): \[\sin(A) = \frac{a \times \sin(B)}{b}\]

Finally, find the area of the triangle using the formula involving two sides and the included angle: \[\text{Area} = \frac{1}{2} \times a \times c \times \sin(B)\] This formula directly uses the two known sides and the included angle to find the area.

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주요 개념

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Law of Cosines

The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles. It is useful for finding an unknown side or angle when two sides and the included angle are known, or when all three sides are known. The formula is c² = a² + b² - 2ab cos(C).

Area of a Triangle Using Two Sides and Included Angle

The area of a triangle can be found using the formula (1/2)ab sin(C), where a and b are two sides and C is the included angle between them. This method is especially useful when the height is not known but two sides and the included angle are given.

Trigonometric Functions and Angle Measurement

Understanding how to use sine and cosine functions with angles measured in degrees is essential. Angles in triangles are typically given in degrees, and trigonometric functions help relate these angles to side lengths, enabling calculation of unknown sides or areas.

In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.B = 125°, a = 8 yards, c = 5 yards