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)

Verified step by step guidance

1. Start by recalling the Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
2. Express \( a \) and \( b \) in terms of \( \sin A \) and \( \sin B \): \( a = k \sin A \) and \( b = k \sin B \), where \( k \) is a constant.
3. Substitute these expressions into the left side of the equation: \( \frac{a - b}{a + b} = \frac{k \sin A - k \sin B}{k \sin A + k \sin B} \).
4. Factor out \( k \) from both the numerator and the denominator: \( \frac{k(\sin A - \sin B)}{k(\sin A + \sin B)} \).
5. Cancel \( k \) from the numerator and the denominator to obtain \( \frac{\sin A - \sin B}{\sin A + \sin B} \), which matches the right side of the equation.

Key Concepts

Law of Sines

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This can be expressed as a/sin(A) = b/sin(B) = c/sin(C). This law is essential for solving problems involving non-right triangles and helps establish relationships between angles and sides.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, angle sum and difference identities, and double angle formulas. These identities are crucial for manipulating and simplifying expressions in trigonometric proofs.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables or prove relationships. In the context of the Law of Sines, it is important to transform the ratios of sides and sines into equivalent forms to demonstrate the validity of the given statement. Mastery of algebraic techniques is essential for effective problem-solving in trigonometry.