A real estate agent wants to find the area of a triangular lot. A surveyor takes measurements and finds that two sides are 52.1 m and 21.3 m, and the angle between them is 42.2°. What is the area of the triangular lot?

검증된 단계별 안내

Identify the given information: two sides of the triangle, \(a = 52.1\) m and \(b = 21.3\) m, and the included angle between them, \(\theta = 42.2^\circ\).

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

Substitute the known values into the formula: \(\text{Area} = \frac{1}{2} \times 52.1 \times 21.3 \times \sin(42.2^\circ)\).

Calculate \(\sin(42.2^\circ)\) using a calculator or trigonometric tables, making sure your calculator is set to degrees.

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.

Triangle Area Using Two Sides and Included Angle

The area of a triangle can be calculated using the formula: (1/2) × side1 × side2 × sin(included angle). This method is useful when two sides and the angle between them are known, allowing direct computation without needing the height.

Sine Function in Trigonometry

The sine function relates an angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. In this context, sine of the included angle helps determine the height component needed to find the area of the triangle.

Angle Measurement in Degrees

Angles can be measured in degrees or radians; here, the angle is given in degrees (42.2°). Understanding how to use degree measures in trigonometric functions is essential for correctly applying formulas and obtaining accurate results.