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Half-angle formulas are trigonometric identities that allow you to express a trigonometric function of half of an angle in terms of the trigonometric functions of the whole angle. These formulas are incredibly useful when dealing with problems where only the half-angle is given or when calculating trigonometric functions indirectly. In our exercise, we make use of such a formula to find \(\sin \theta\) when we know \(\tan(\theta/2)\). The specific half-angle formula we use is:

• \(\sin \theta = \frac{2 \tan(\theta/2)}{1 + \tan^2(\theta/2)}\)

This formula helps us calculate the sine of the fuller angle \(\theta\) by utilizing the tangent of half that angle, \(\theta/2\). By substituting \(\tan(\theta/2) = -\pi/3\) into this identity, we can work through to find \(\sin \theta\). Such formulas are particularly powerful because they simplify complex calculations by using known values effectively. Understanding these relationships provides a deeper grasp of trigonometric functions and their myriad interconnections.

The sine function, one of the primary trigonometric functions, is a measure of the y-coordinate of a point on the unit circle as it relates to the angle. Specifically, for an angle \(\theta\), \(\sin \theta\) represents the perpendicular height from the x-axis to the radius that forms the angle, given a circle with a radius of one. The value of \(\sin \theta\) varies between -1 and 1 as \(\theta\) sweeps through different quadrants:

• In the first and second quadrants, \(\sin \theta\) is positive.
• In the third and fourth quadrants, \(\sin \theta\) is negative.

In our exercise, \(\theta\) is located in the third quadrant (\(180^\circ < \theta < 270^\circ\)), where sine values are negative. This aligns with our solution, where a negative sine was maintained. Consistently understanding the sine values in different quadrants is crucial as it influences the outcome heavily, such as the sign of the result in calculations. Hence, the knowledge of the sine function, its behavior, and its relation to the unit circle is essential in mastering trigonometry.

The tangent function is another fundamental trigonometric function that relates to the angle in a clockwise circle. Defined as the ratio of the sine function to the cosine function:

• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

It provides a unique perspective on angles as it measures the slope of the radius that forms \(\theta\) against the x-axis. The values of the tangent function can extend from negative to positive infinity, reflecting the steepness of the slope:

• Tangent is positive in the first and third quadrants.
• Tangent is negative in the second and fourth quadrants.

In this context, since we’re dealing with \(\tan(\theta/2)\), knowing \(\tan(\theta/2) = -\pi/3\) positioned \(\theta/2\) in a quadrant where tangent is negative. This understanding aligns with facile navigation through half-angle problems, aiding in piecing together larger trigonometric puzzles. Understanding how to transition from \(\tan(\theta/2)\) to \(\sin \theta\) highlights the interplay between trigonometric functions, proving useful across mathematical and real-world scenarios.