In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.
A = 22°, b = 20 feet, c = 50 feet

검증된 단계별 안내

Identify the given elements of the triangle: angle \(A = 22^\circ\), side \(b = 20\) feet, and side \(c = 50\) feet. We need to find the area of the triangle.

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: \(\text{Area} = \frac{1}{2} \times 20 \times 50 \times \sin 22^\circ\).

Calculate \(\sin 22^\circ\) using a calculator or trigonometric table to find the sine of the angle.

Multiply the values together to find the area, then round the result to the nearest whole number as requested.

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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 unknown sides or angles in any triangle, especially when two sides and the included angle 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 calculated using two sides and the included angle with the formula: Area = 1/2 * b * c * sin(A). This method is efficient when two sides and the angle between them are given, avoiding the need to find the third side.

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 are fundamental in solving oblique triangles and calculating areas.

In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.A = 22°, b = 20 feet, c = 50 feet