Solving Trigonometric Equations: sin4x - sin3x 0

When faced with solving a trigonometric equation like sin4x - sin3x 0, the application of trigonometric identities can simplify the process significantly. This article will guide you through the steps to solve the equation and discuss the underlying principles and methods used in trigonometric problem-solving.

Understanding the Basics of Trigonometric Identities

In trigonometry, identities are equations involving trigonometric functions that are true for all values input into the function, as long as they occur within the domain of the functions. One of the fundamental identities is the sum-to-product identity for sine, which states:

sin(a) - sin(b) 2 cos left( frac{a b}{2}right) sin left( frac{a - b}{2}right)

Applying the Sum-to-Product Identity

The given equation is sin4x - sin3x 0. By using the sum-to-product identity, we can rewrite this equation as:

sin4x - sin3x 2 cos left( frac{4x 3x}{2}right) sin left( frac{4x - 3x}{2}right) 0

Further simplifying, we get:

2 cos left( frac{7x}{2}right) sin left( frac{x}{2}right) 0

Setting Each Factor to Zero

The product of two terms equals zero if and only if at least one of the terms is zero. Therefore, we have two separate equations to solve:

Solving (cos left( frac{7x}{2}right) 0)

This equation holds if:

frac{7x}{2} frac{pi}{2} npi

Multiplying both sides by 2:

7x pi 2npi

Dividing by 7:

x frac{pi}{7} frac{2npi}{7}

Solving (sin left( frac{x}{2}right) 0)

This equation holds if:

frac{x}{2} kpi

Multiplying both sides by 2:

x 2kpi

Combining the Solutions

The complete solution to the equation sin4x - sin3x 0 is the union of the solutions to the two separate equations. Therefore, the solutions are:

Conclusion

Thus, solving the trigonometric equation sin4x - sin3x 0 involves identifying the zeros of both the sine and cosine functions within the transformed equation. The methods used here—sum-to-product identities and the solving of simple trigonometric equations—are key tools in the arsenal of any student or professional working with trigonometric equations.