91Ó°ÊÓ

Understanding the identities for sin (x+y) and sin (x-y) is essential for simplifying expressions and solving trigonometric equations. These identities are particularly useful when dealing with angles that are sums or differences of two known angles.

When we expand sin (x+y), we use the identity:
\[ \sin (x+y) = \sin x \cos y + \cos x \sin y \]
This formula is derived from the angle sum property of the sine function. Similarly, for sin (x-y), the expansion is:
\[ \sin (x-y) = \sin x \cos y - \cos x \sin y \]
These expansions allow us to transform the sum of angles into a product of sines and cosines. The advantage? It often simplifies the expressions and thus our calculations. For instance, in our original exercise, the formulae provided a straightforward path to verify the given identity.

The process of verifying trigonometric identities might seem daunting, but it is essentially about transforming one side of an equation, so it matches the other side. To do this, we often employ expansion formulas, like the ones for sine and cosine, and algebraic manipulation.

The goal is to show that both sides are equal for all values within the domains of the variables involved. Here's a simplified roadmap to success:

• Start with the more complex side of the identity and try to simplify it.
• Use known trigonometric identities to rewrite the terms.
• Look for common factors, opportunities to combine terms, and techniques to condense expressions.
• Keep simplifying until both sides of the equation look the same.

While verifying the identity in our exercise, we took the more complex expression and used expansion identities to break it down. In the end, both sides of the equation matched, thereby verifying the given identity.

The sine and cosine expansion formulas are pivotal in trigonometry. They help us dissect and reconfigure problems involving angles, especially those that are not standard angles found on the unit circle.

The expansion formulas are:

Sine Expansion:

• \( \sin(x \pm y) = \sin x \cos y \pm \cos x \sin y \)

Cosine Expansion:

• \( \cos(x \pm y) = \cos x \cos y \mp \sin x \sin y \)

These formulas allow us to express the sine or cosine of a sum or difference of angles in terms of the products of sines and cosines of those angles.

Expansion formulas are particularly helpful when angles are not readily in a form that can be easily evaluated, such as non-special angles. They are one of the primary tools for solving trigonometric equations, simplifying expressions, and verifying identities - as we have practiced in our example problem.