Find the area of each triangle ABC.

A = 56.80°, b = 32.67 in., c = 52.89 in.

검증된 단계별 안내

Identify the given elements: angle \(A = 56.80^\circ\), side \(b = 32.67\) inches, and side \(c = 52.89\) inches. We need to find the area of triangle \(ABC\).

Recall the formula for the area of a triangle when two sides and the included angle are known: \(\text{Area} = \frac{1}{2}bc \sin A\).

Substitute the known values into the formula: \(\text{Area} = \frac{1}{2} \times 32.67 \times 52.89 \times \sin(56.80^\circ)\).

Calculate \(\sin(56.80^\circ)\) using a calculator or trigonometric table, ensuring your calculator is in degree mode.

Multiply the values together to find the area: first multiply the two sides, then multiply by the sine of the angle, and finally multiply by \(\frac{1}{2}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Law of Cosines

The Law of Cosines relates the lengths of the 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. The formula is c² = a² + b² - 2ab cos(C), which helps in determining missing elements in oblique triangles.

Area of a Triangle Using Two Sides and Included Angle

The area of a triangle can be calculated using the formula (1/2)bc sin(A), where b and c are two sides and A is the included angle between them. This method is especially useful when the height is unknown but two sides and the included angle are given, allowing direct computation of the area.

Trigonometric Functions and Angle Measurement

Understanding sine and cosine functions and how to apply them to angles measured in degrees is essential. These functions relate angles to ratios of sides in right triangles and extend to oblique triangles, enabling calculation of unknown sides or areas using given angles and side lengths.

Find the area of each triangle ABC.A = 56.80°, b = 32.67 in., c = 52.89 in.