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The sine function is a fundamental concept in trigonometry. It measures the y-coordinate of a point on a unit circle, which is a circle with a radius of 1 unit, as it rotates around its center. For any angle \( \theta \), the sine value ranges from -1 to 1.

The sine of an angle is periodic with a period of \( 360^{\circ} \), meaning it repeats its values every \( 360^{\circ} \). This function is useful for modeling various oscillatory phenomena like sound waves or tides. Understanding the sine function helps in solving equations like \( \sin 2\theta = \frac{\sqrt{3}}{2} \), where we seek specific angles within a defined range for which the sine value matches a given number.

Some properties to remember about the sine function:

• It is an odd function, implying \( \sin(-\theta) = -\sin(\theta) \).
• The graph of the sine function is a smooth wave that extends indefinitely in both directions.

When tasked with solving for angles corresponding to a specific sine value, it is essential to leverage known reference angles and the periodic nature of trigonometric functions. With \( \sin 2\theta = \frac{\sqrt{3}}{2} \), we recognize this sine value from common angles such as \( 60^{\circ} \) and \( 120^{\circ} \).

Since we are solving for \( 2\theta \), we calculate the possible values as multiples of these reference angles:

• \( 2\theta = 60^{\circ} \) and \( 2\theta = 120^{\circ} \)
• Accounting for periodicity, \( 2\theta = 60^{\circ} + 360^{\circ}k \) and \( 2\theta = 120^{\circ} + 360^{\circ}k \) where \( k \) is an integer.

This periodic adjustment ensures no solutions are missed. Dividing each result by 2 gives the solutions for \( \theta \), leading us to angles \( 30^{\circ}, 60^{\circ}, 210^{\circ} \), and \( 240^{\circ} \). These solutions fall within the initial \( 0^{\circ} \leq \theta < 360^{\circ} \) range.

Graphical verification is a powerful method for confirming analytical solutions in trigonometry. By plotting the functions \( y = \sin 2\theta \) and \( y = \frac{\sqrt{3}}{2} \) on the same graph and observing their intersections, we can visually check our calculated angle solutions.

The graph of \( \sin 2\theta \) will show a wave that repeats every \( 180^{\circ} \), due to the doubled angle inside the sine function. It results in two full oscillations between 0 and \( 360^{\circ} \). The line \( y = \frac{\sqrt{3}}{2} \) is a horizontal line intersecting the sine wave at specific points corresponding to the solutions we found analytically.

By marking the intersections, we confirm that they occur at \( \theta = 30^{\circ}, 60^{\circ}, 210^{\circ}, \) and \( 240^{\circ} \). This graphical approach not only verifies our work but also provides a visual understanding of how the trigonometric functions interact within the given domain.