Solve the following equations.
$\(\sin\)3x=\(\frac{\sqrt{2}\)}{2},0\(\leq{x}\]\lt{2\pi}\)$

Verified step by step guidance

1. First, recognize that the equation \( \sin(3x) = \frac{\sqrt{2}}{2} \) is a trigonometric equation where we need to find the values of \( x \) within the interval \( 0 \leq x < 2\pi \).
2. Recall that \( \sin(\theta) = \frac{\sqrt{2}}{2} \) corresponds to angles \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \) in the unit circle.
3. Since we have \( \sin(3x) = \frac{\sqrt{2}}{2} \), set \( 3x = \frac{\pi}{4} + 2k\pi \) and \( 3x = \frac{3\pi}{4} + 2k\pi \) for integer values of \( k \).
4. Solve for \( x \) by dividing each equation by 3: \( x = \frac{\pi}{12} + \frac{2k\pi}{3} \) and \( x = \frac{\pi}{4} + \frac{2k\pi}{3} \).
5. Determine the values of \( k \) such that \( x \) remains within the interval \( 0 \leq x < 2\pi \). Calculate these values to find all possible solutions for \( x \).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to the ratios of sides in right triangles. The sine function, specifically, gives the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is crucial for solving equations involving angles, especially in the context of periodic functions like sine, which repeat their values in regular intervals.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsine, allow us to determine the angle that corresponds to a given sine value. For example, if \\sin(x) = y, then \\arcsin(y) = x. These functions are essential for solving equations where the angle is unknown, particularly when working with equations like \\sin(3x) = \\frac{\sqrt{2}}{2}, as they help find the specific angle solutions within a defined range.

Periodic Nature of Sine Function

The sine function is periodic with a period of 2π, meaning it repeats its values every 2π radians. This property is important when solving trigonometric equations, as it implies that there can be multiple solutions within a specified interval. For the equation \\sin(3x) = \\frac{\sqrt{2}}{2}, recognizing the periodicity allows us to find all possible values of x that satisfy the equation within the interval \\[0, 2π)\].