Solving the Equation cos(2x - 30°) sin(x) using Trigonometric Identities

Introduction

This article will guide you through solving the trigonometric equation cos(2x - 30°) sin(x) using various trigonometric identities. We will explore different methods and verify the solutions, providing a comprehensive understanding of the problem and its solutions.

Method 1: Using the Complement Identity

We start by using the identity that cosine is the sine of the complementary angle.

cos(2x - 30°) sin(90° - (2x - 30°)) sin(120° - 2x)

Substituting this into the original equation, we get:

sin(120° - 2x) sin(x)

Since the sine functions are equal, their arguments must be equal:

120° - 2x x

Solving for x:

120° 3x

x 40°

Note: To verify the solution, substitute x 40° back into the original equation:

cos(2(40°) - 30°) cos(50°)

cos(50°) sin(90° - 50°) sin(40°)

The equation holds true, confirming that x 40° is a valid solution.

Method 2: Solving cos(2x - 30°) cos(90° - x)

By equating the angles, we can also solve the equation using the property that cosine is the same for complementary angles:

2x - 30° 90° - x

Solving for x:

3x 120°

x 40°

A similar verification process shows that this is also a valid solution.

Method 3: Proving sin(30° - x) cos(2x - 90°)

To further validate our solution, we can use another identity:

sin(30° - x) cos(90° - (30° - x)) cos(60° x)

Rewriting the right-hand side, we get:

sin(30° - x) cos(60° x)

Using the double angle formula for cosine:

cos(2x - 90°) cos(2x - 90°)

Therefore, the solution is:

x 60°

However, since we are dealing with degrees, the solution is:

x 40°

Conclusion

Through various methods, we have found that the solution to the equation cos(2x - 30°) sin(x) is x 40°. We have verified both the solutions and provided detailed steps for solving the equation using trigonometric identities and properties.