Solving the Equation sinA sin2A sin3A 0 Using Trigonometric Identities

In this article, we will explore how to solve the trigonometric equation sin A sin 2A sin 3A 0 using trigonometric identities. We will break down the steps in a detailed manner to ensure a thorough understanding of the process.

Introduction to Trigonometric Identities

Trigonometric identities are mathematical equations that are true for every value of the variable involved. They are essential in simplifying and solving complex trigonometric equations. In this article, we will utilize some key identities to solve the given problem.

Step-by-Step Solution

Step 1: Utilize Double-Angle Identity

First, we start with the double-angle identity for sine, which states:

sin 2A 2 sin A cos A

Substituting this identity into the original equation:

sin A sin 2A sin 3A 0 Rightarrow sin A (2 sin A cos A) sin 3A 0 Rightarrow 2 sin^2 A cos A sin 3A 0

Step 2: Utilize Triple-Angle Identity

Next, we use the triple-angle identity for sine, which is:

sin 3A 3 sin A - 4 sin^3 A

Substituting this identity into the previous equation:

2 sin^2 A cos A (3 sin A - 4 sin^3 A) 0 Rightarrow 2 sin^2 A cos A cdot 3 sin A - 2 sin^2 A cos A cdot 4 sin^3 A 0 Rightarrow 6 sin^3 A cos A - 8 sin^5 A cos A 0 Rightarrow 2 sin^3 A (3 - 4 sin^2 A) cos A 0

Step 3: Solving the Equation

The equation can be satisfied in two ways:

sin A 0, which implies A npi where n is an integer. 3 - 4 sin^2 A 0, which implies sin^2 A frac{3}{4}.

Case 2: Further Analysis

For the second case, solving for sin^2 A frac{3}{4} leads to:

sin A pm sqrt{frac{3}{4}} pm frac{sqrt{3}}{2}

Given that the sine function is bounded between -1 and 1, the value frac{sqrt{3}}{2} is valid, and we get:

sin A frac{sqrt{3}}{2} Rightarrow A npi (-1)^n frac{pi}{3}

sin A -frac{sqrt{3}}{2} Rightarrow A npi (-1)^n left(pi - frac{pi}{3}right) npi - (-1)^n frac{pi}{3}

Therefore, the solutions to the equation sin A sin 2A sin 3A 0 are:

A npi quad text{or} quad A npi (-1)^n frac{pi}{3} quad text{or} quad A npi - (-1)^n frac{pi}{3}

Conclusion

Through the use of trigonometric identities, we have successfully solved the equation sin A sin 2A sin 3A 0. The solutions include multiples of pi, values that satisfy sin A frac{sqrt{3}}{2}, and values that satisfy sin A -frac{sqrt{3}}{2}.

Understanding these steps and the relevant trigonometric identities is crucial for solving more complex trigonometric equations and for further exploration in trigonometry and related fields such as calculus and physics.