91Ó°ÊÓ

The addition formula is crucial for expanding trigonometric expressions, especially when dealing with terms like \( \sin(3\theta) \). This formula helps in breaking down a complex sine or cosine of sums into more manageable pieces.
For sine, the addition formula is given by:

• \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)

In our exercise, we use it to expand \( \sin(2\theta + \theta) \) which becomes \( \sin(2\theta)\cos(\theta) + \cos(2\theta)\sin(\theta) \). This helps transform the original sine of a sum into products of sine and cosine which can be simplified further.
Understanding how to use the addition formula allows us to manipulate and simplify expressions involving multiple angles, setting the stage for further use of other formulas.

The double angle formula is another important tool in trigonometry used to simplify expressions involving double angles. It lets us express trigonometric functions of \( 2\theta \) in terms of a single angle \( \theta \).
For instance, the double angle formulas are:

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)
• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)

In this exercise, we apply these formulas to \( \sin(3\theta) = \sin(2\theta)\cos(\theta) + \cos(2\theta)\sin(\theta) \). By substituting the double angle formulas, we simplify the equation significantly.
This simplification process reduces the expression to products and powers of sine and cosine which can further be solved using other identities.

The Pythagorean identity is fundamental in trigonometry as it establishes a relationship between the sine and cosine of the same angle. This identity states:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)

We use the Pythagorean identity to replace \( \cos^2(\theta) \) with \( 1 - \sin^2(\theta) \) in the equation during our problem-solving process.
This substitution is crucial for simplifying terms involving \( \cos^2(\theta) \). For example, turning \( 2\sin(\theta)\cos^2(\theta) \) into \( 2\sin(\theta)(1 - \sin^2(\theta)) \) makes it easier to further reduce and rearrange the equation.
The Pythagorean identity is thus a powerful tool for transforming expressions and achieving the desired form, such as \( 3\sin(\theta) - 4\sin^3(\theta) \) in our final expression.