Trigonometry is a fundamental branch of mathematics that explores the relationships between angles and sides of triangles. It plays a crucial role in various fields, including physics, engineering, and computer science. One of the key concepts in trigonometry is simplifying trigonometric expressions. This article delves into the simplification of the expression $\sin^4(x)$, providing a step-by-step approach and highlighting the underlying principles.
Simplifying $\sin^4(x)$
The expression $\sin^4(x)$ represents the fourth power of the sine function evaluated at angle x. To simplify this expression, we can utilize trigonometric identities and algebraic manipulations. Here's a detailed breakdown of the process:
1. Expressing the Fourth Power as a Square
We begin by recognizing that $\sin^4(x)$ can be written as the square of $\sin^2(x)$:
$\sin^4(x) = (\sin^2(x))^2$
2. Applying the Double-Angle Formula
The double-angle formula for cosine is a powerful tool for simplifying trigonometric expressions involving squares. It states:
$\cos(2x) = 1 - 2\sin^2(x)$
Solving this equation for $\sin^2(x)$, we get:
$\sin^2(x) = \frac{1 - \cos(2x)}{2}$
3. Substituting the Double-Angle Formula
Substituting this expression for $\sin^2(x)$ into our original expression, we get:
$\sin^4(x) = \left(\frac{1 - \cos(2x)}{2}\right)^2$
4. Expanding the Square
Expanding the square on the right-hand side, we obtain:
$\sin^4(x) = \frac{1 - 2\cos(2x) + \cos^2(2x)}{4}$
5. Applying the Double-Angle Formula Again
We can further simplify this expression by using the double-angle formula for cosine again:
$\cos(4x) = 2\cos^2(2x) - 1$
Solving for $\cos^2(2x)$, we get:
$\cos^2(2x) = \frac{\cos(4x) + 1}{2}$
6. Substituting and Simplifying
Substituting this expression for $\cos^2(2x)$ into our equation, we get:
$\sin^4(x) = \frac{1 - 2\cos(2x) + \frac{\cos(4x) + 1}{2}}{4}$
Simplifying the numerator:
$\sin^4(x) = \frac{3 - 4\cos(2x) + \cos(4x)}{8}$
7. Final Simplified Expression
Therefore, the simplified expression for $\sin^4(x)$ is:
$\sin^4(x) = \frac{3 - 4\cos(2x) + \cos(4x)}{8}$
This expression is now in a form that is more manageable and easier to work with in various trigonometric calculations.
Conclusion
Simplifying trigonometric expressions like $\sin^4(x)$ involves applying a combination of trigonometric identities and algebraic manipulations. By using the double-angle formula for cosine and substituting appropriately, we can reduce the expression to a more compact and manageable form. Understanding these simplification techniques is crucial for solving trigonometric equations, analyzing wave functions, and tackling various problems in fields that rely on trigonometric concepts.
