The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then

(sin A) / a = (sin B) / b = (sin C) / c

Use the accompanying figures and the identity sin (π − θ) = sin θ, if required, to derive the law.

검증된 단계별 안내

Start by considering a triangle with sides a, b, and c opposite to angles A, B, and C respectively. The goal is to relate the sides and angles using the law of sines.

Recall the formula for the area of a triangle: Area = (1/2) * base * height. For triangle ABC, you can express the area using each side as the base. For example, using side a as the base, the height is b * sin(C), so the area is (1/2) * a * b * sin(C).

Similarly, express the area using side b as the base: Area = (1/2) * b * c * sin(A), and using side c as the base: Area = (1/2) * c * a * sin(B).

Since all these expressions represent the same area, set them equal to each other: (1/2) * a * b * sin(C) = (1/2) * b * c * sin(A) = (1/2) * c * a * sin(B).

Divide each part of the equation by the product of the sides involved to isolate the sine terms: (sin A) / a = (sin B) / b = (sin C) / c. This is the law of sines, which shows the relationship between the sides and angles of a triangle.

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주요 개념

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

Law of Sines

The Law of Sines is a fundamental relationship in trigonometry that relates the angles and sides of a triangle. It states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles. This can be expressed mathematically as (sin A) / a = (sin B) / b = (sin C) / c, where A, B, and C are the angles, and a, b, and c are the lengths of the sides opposite those angles.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One important identity is sin(π - θ) = sin θ, which indicates that the sine function is symmetric about π/2. This identity can be useful in deriving relationships in triangles, particularly when working with angles that sum to π or 180 degrees.

Triangle Properties

Understanding the properties of triangles is essential for applying the Law of Sines. A triangle's angles always sum to 180 degrees, and the relationship between the angles and sides is crucial for solving problems. Additionally, knowing how to identify and label the sides and angles correctly is vital for applying the Law of Sines effectively in various geometric contexts.

The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then(sin A) / a = (sin B) / b = (sin C) / cUse the accompanying figures and the identity sin (π − θ) = sin θ, if required, to derive the law.