91Ó°ÊÓ

In trigonometry, the double-angle formula is a crucial identity that relates the trigonometric functions of double angles to the functions of single angles. For sine, the double-angle formula is:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
However, it can also be expressed using tangent, which is especially useful in specific contexts, like the given problem. The version we use here is:
\[ \sin(\theta) = \frac{2 \tan(\frac{\theta}{2})}{1 + \tan^2(\frac{\theta}{2})} \]
Understanding this form is essential when the problem gives you a tangent instead of sine or cosine directly. You can convert the problem into simpler parts by using this powerful identity.

The tangent half-angle identity is a handy tool in trigonometry, especially when dealing with problems involving half-angles. The identity for the tangent of half an angle \( \frac{\alpha}{2} \) is given by:
\[ \tan \frac{\alpha}{2} = z \]
In our context, we use this identity to express other trigonometric functions in terms of \( z \). For sine, the formula evolves into:
\[ \sin(\alpha) = \frac{2z}{1 + z^2} \]
This transformation leverages the tangent half-angle to simplify the problem and reach a solution quicker. A good grasp of this identity not only helps in examinations but also builds a stronger understanding of trigonometric relationships.

The substitution method is a mathematical technique used to simplify problems by replacing variables with their given values or expressions. In our problem, we use the substitution method to make the given trigonometric identities fit the form we need.
We start with the given condition: \( z = \tan \frac{\alpha}{2} \).
We then substitute \( z \) into the double-angle formula:
\[ \sin(\alpha) = \frac{2 \tan (\frac{\alpha}{2}) }{ 1 + \tan^2 (\frac{\alpha}{2}) } \]
Given \( z = \tan (\frac{\alpha}{2}) \), the formula becomes:
\[ \sin(\alpha) = \frac{2z}{1 + z^2} \]
Using substitution simplifies the problem by breaking it down into smaller, more manageable parts. This method is powerful for tackling complex equations and systems.