Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.

(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)

검증된 단계별 안내

Recall the Law of Sines, which states that for any triangle ABC with sides a, b, and c opposite angles A, B, and C respectively, the following holds: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\] where $R$ is the radius of the triangle's circumscribed circle.

From the Law of Sines, express sides $a$ and $b$ in terms of $\sin A$ and $\sin B$: \[a = 2R \sin A\] \[b = 2R \sin B\]

Substitute these expressions for $a$ and $b$ into the left side of the equation to be proved: \[\frac{a - b}{a + b} = \frac{2R \sin A - 2R \sin B}{2R \sin A + 2R \sin B}\]

Factor out \$2R$ from numerator and denominator: \[\frac{2R (\sin A - \sin B)}{2R (\sin A + \sin B)}\] Since $2R$ is common in numerator and denominator, it cancels out, leaving: \[\frac{\sin A - \sin B}{\sin A + \sin B}\]

This shows that \[\frac{a - b}{a + b} = \frac{\sin A - \sin B}{\sin A + \sin B}\] which completes the proof using the Law of Sines.

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주요 개념

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Law of Sines

The Law of Sines states that in any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C. This relationship allows us to connect side lengths and angles, making it essential for proving identities involving sides and sines of angles.

Algebraic Manipulation of Ratios

Understanding how to manipulate ratios and fractions is crucial for transforming expressions like (a - b)/(a + b) and (sin A - sin B)/(sin A + sin B). This involves factoring, cross-multiplying, and simplifying terms to show equivalence between two ratios.

Properties of Sine Function in Triangles

The sine function relates angles to side lengths in triangles. Recognizing how sine values change with angles and how differences and sums of sines behave helps in comparing expressions like (sin A - sin B) and (sin A + sin B), which is key to proving the given identity.

Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)