Solving the Equation sin(2x) √3/2 and Understanding Its Implications

Understanding the Problem: The problem at hand is to solve the equation sin(2x) √3/2. This involves understanding the properties of the sine function and its periodicity. To solve this, we need to take a step-by-step approach, starting from recognizing the angles where the sine of an angle equals √3/2.

Recognizing Key Angles

Two angles that have a sine value of √3/2 are π/3 (or 60 degrees) and 2π/3 (or 120 degrees). These angles are derived from the properties of a 30-60-90 triangle, where the sine of 60 degrees is √3/2.

Unit Circle Visualization: A unit circle can be used to visualize these angles. Points corresponding to these angles have a y-coordinate (sin value) of √3/2. By extending the periodicity of the sine function, we can find all possible solutions to the equation.

General Solution

Starting from the equation:

sin(2x) √3/2

We can express:

2x arcsin(√3/2) 2πn

Where n is any integer. This is because the sine function is periodic with a period of 2π. The arcsin function gives us the principal value satisfying the equation.

Therefore, solving for x, we get:

x 1/2 * arcsin(√3/2) πn

Since we know that arcsin(√3/2) π/3, we can substitute this value in:

x 1/2 * π/3 πn

Which simplifies to:

x π/6 πn

Specific Solutions within 0 ≤ x ≤ 2π

Within the specific interval from 0 ≤ x ≤ 2π, we have:

x π/6 x π/3 x 7π/6 x 4π/3

These solutions can be found by substituting n 0, 1, 2, 3 respectively.

Verifying with a Graphical Approach

To verify the solutions, we can plot the function y sin(2x) - √3/2. The points where this function equals zero will give us the solutions to the equation. By plotting this, we can see the zeros align with the angles we have calculated.

Trigonometric Facts and Relationships

Here are some key trigonometric facts related to this problem:

sin(60°) √3/2 sin(π/3 radians) cos(60°) 1/2 sin(30°) 1/2 cos(60°) cos(30°) √3/2 sin(60°)

These relationships can be used for quick reference or when solving similar trigonometric equations.

Conclusion

By understanding the properties of sine and using the unit circle, we can systematically solve equations like sin(2x) √3/2. This problem not only enhances one's trigonometric skills but also reinforces the periodic nature of trigonometric functions.

Related Keywords

Sin(2x), solving trigonometric equations, unit circle