Solving Trigonometric Equations with Sum-to-Product Identities
Introduction
Trigonometric equations are a fundamental part of mathematics used in various fields, including physics, engineering, and computer science. This article will guide you through the process of solving the equation $sin 2x - sin 4x - sin 6x 0$ using sum-to-product identities. These identities are valuable for simplifying and solving complicated trigonometric expressions.
Step-by-Step Solution
Let's begin by rewriting the given equation using Sum-to-Product Identities. The identities we will use are:
$sin A - sin B 2 cosleft(frac{A B}{2}right) sinleft(frac{A - B}{2}right)$The equation is:
$sin 2x - sin 4x - sin 6x 0$
We can rearrange this equation as:
$sin 2x sin 6x sin 4x$
Using the Sum-to-Product Identity, we rewrite $sin 2x sin 6x$:
$sin 2x sin 6x 2 sinleft(frac{2x 6x}{2}right) cosleft(frac{6x - 2x}{2}right) 2 sin 4x cos 2x$
Substituting this back into the original equation, we get:
$2 sin 4x cos 2x sin 4x$
Next, we rearrange this equation to:
$2 sin 4x cos 2x - sin 4x 0$
Factoring out $sin 4x$, we obtain:
$sin 4x (2 cos 2x - 1) 0$
This gives us two cases to consider:
Case 1: $sin 4x 0$
The solutions for $sin 4x 0$ are:
$4x npi$ for $n in mathbb{Z}$
Thus,
$x frac{npi}{4}$ for $n in mathbb{Z}$
Case 2: $2 cos 2x - 1 0$
Rearranging this equation, we get:
$cos 2x frac{1}{2}$
The solutions for $cos 2x frac{1}{2}$ are:
$2x 2kpi pm frac{pi}{3}$ for $k in mathbb{Z}$
Thus,
$x frac{6kpi pm pi}{6}$ for $k in mathbb{Z}$
Conclusion
The complete solution set is given by:
$x frac{npi}{4}$ for $n in mathbb{Z}$ $x frac{6kpi pm pi}{6}$ for $k in mathbb{Z}$By understanding and applying the sum-to-product identities, solving trigonometric equations like this one becomes much more manageable. Trigonometric identities are powerful tools in algebra and calculus, enhancing problem-solving skills in mathematics.