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The sine function is a fundamental trigonometric function that measures the y-coordinate of a point on the unit circle corresponding to a given angle. The function is periodic, meaning it repeats its values in regular intervals. Specifically, it has a period of \(2\pi\), which indicates that \( \sin(\theta) = \sin(\theta + 2\pi k) \) for any integer \(k\). For example:

• \( \sin(0) = 0 \)
• \( \sin(\pi/2) = 1 \)
• \( \sin(\pi) = 0 \)
• \( \sin(3\pi/2) = -1 \)
• \( \sin(2\pi) = 0 \)

When dealing with negative angles, recall that sine is an odd function, which means \( \sin(-\theta) = - \sin(\theta) \). This property can help simplify expressions with negative angles and was used in evaluating \( \sin(-\pi/2) \) in the given exercise.

The cosine function is another key trigonometric function that measures the x-coordinate of a point on the unit circle for a given angle. It also is periodic, with a period of \(2\pi\). The periodicity indicates that \( \cos(\theta) = \cos(\theta + 2\pi k) \) for any integer \(k\). Examples include:

• \( \cos(0) = 1 \)
• \( \cos(\pi/2) = 0 \)
• \( \cos(\pi) = -1 \)
• \( \cos(3\pi/2) = 0 \)
• \( \cos(2\pi) = 1 \)

For negative angles, remember that cosine is an even function, meaning \( \cos(-\theta) = \cos(\theta) \). This property simplifies expressions involving negative angles and was relevant when addressing \( \cos(-\pi/6) \) and \( \cos(-\pi/3) \) in the exercise.

The angle addition formula is a powerful trigonometric identity that simplifies the calculation of sine or cosine for the sum of two angles. The formula for sine is:

\( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \).

This identity allows us to express the sine of a sum as the sum of products of sines and cosines of the individual angles. It is widely used in solving trigonometric equations and in simplifying expressions. For the given exercise:

\( \sin(-\pi/6)\cos(-\pi/3) + \cos(-\pi/6)\sin(-\pi/3) \),
we applied the formula directly:

\( \sin(-\pi/6 + -\pi/3) = \sin(-\pi/2) \).

Knowing the value of \( \sin(-\pi/2) = -1 \), we could readily find the simplified result by combining angle measures and evaluating the sine function at the result.