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The sum of angles formula is a key trigonometric identity that helps in expanding trigonometric functions involving sums of multiple angles. Specifically, the formula for sine is: \( \sin(X + Y) = \sin X \cos Y + \cos X \sin Y \). This formula is essential when dealing with angles expressed as the sum of two or more different angles.

To solve complex problems, like the one in our exercise where \( \sin(A + B + C) \) needs to be expanded, we can apply the sum of angles formula multiple times. Here's a brief look at the approach:

• Break down \( A + B + C \) into two parts, such as \( A \) and \( B + C \).
• Apply the formula \( \sin(A + (B+C)) \) by treating it as \( \sin(A + Y) \) where \( Y = B + C \).
• This results in two separate components: \( \sin A \cos(B + C) + \cos A \sin(B + C) \).
• Further expand \( \cos(B+C) \) and \( \sin(B+C) \) using their respective sum formulas.

By structuring the problem into more manageable parts, utilizing these formulas becomes more intuitive and, ultimately, leads to verifying complex identities.

The sine function is one of the primary trigonometric functions and is essential for understanding relationships in a right-angled triangle. For any angle \( \theta \), \( \sin(\theta) \) is defined as the ratio of the length of the side opposite the angle to the hypotenuse.

In trigonometric expansion, such as the expansion of \( \sin(A + B + C) \), the sine function plays a pivotal role. It's involved directly when applying the sum of angles formula and helps in examining how the combined effect of angles impacts trigonometric values. Here’s how:

• When expanding \( \sin(A + B + C) \) into smaller components, the sine function is used to express the contributions of each angle separately.
• For example, \( \sin(A + B + C) \) when initially expanded, includes \( \sin A \) as a multiplicative component of \( \cos(B + C) \).
• Further breakdown leads to combinations involving \( \sin B \) and \( \sin C \), each reflecting contributions from those angles.

This function, because of its versatility and prevalence in trigonometry, allows insights into angle dynamics, making it indispensable for verifying identities like the one given.

Trigonometric expansion is the process of breaking down complex trigonometric expressions into simpler components, often making verification of identities feasible. In our exercise, \( \sin(A + B + C) \) is fully expanded using known trigonometric identities for sine and cosine.

The primary steps in trigonometric expansion, especially for three-angle sums, involve:

• Applying the sum of angles formula for \( \sin(A + (B+C)) \) and simplifying the expression into basic components.
• Breaking down \( \cos(B+C) \) and \( \sin(B+C) \) into their constituent terms using their respective formulas \( \cos(B+C) = \cos B \cos C - \sin B \sin C \) and \( \sin(B+C) = \sin B \cos C + \cos B \sin C \).
• Substituting these back into the combined expression derived earlier, allowing the terms to clearly correlate with the identity to be verified.

This multi-layered process of expansion not only simplifies expressions but clarifies the intricate relationships between angles, enhancing one's ability to authenticate complicated trigonometric identities.