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The angle addition formula is a fundamental identity in trigonometry that helps calculate the trigonometric function of an angle made by adding two angles together. It's particularly important because it breaks down a more complex angle into simpler components associated with special angles and trigonometric functions like sine and cosine. For the cosine function, the angle addition formula is given by:

• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

This formula allows us to express the cosine of the sum of two angles \( a \) and \( b \) in terms of the cosines and sines of these angles.
In our exercise with \( \cos \left(\frac{3 \pi}{2} + x\right) \), \(a\) is \(\frac{3 \pi}{2}\) and \(b\) is \(x\). Understanding how to apply this formula correctly is crucial for solving trigonometric problems efficiently.

The unit circle is a vital concept in trigonometry, serving as a comprehensive tool in understanding angles and their corresponding sine and cosine values. A circle with a radius of 1, it is centered at the origin on a coordinate plane.
Angles on the unit circle are typically measured in radians, which provide a direct relationship between the arc length on the unit circle and the angle itself. As angles are measured counter-clockwise from the positive x-axis, various known points on the unit circle correspond to specific angle measures and their trigonometric values.

• \( \cos \frac{3 \pi}{2} = 0 \)
• \( \sin \frac{3 \pi}{2} = -1 \)

These values come directly from the coordinates of points on the unit circle. For \( \frac{3 \pi}{2} \), which is 270 degrees, the point lies on the negative y-axis. Knowing these coordinates facilitates the calculation and application of trigonometric identities like the angle addition formula.

Special angles in trigonometry, such as multiples of \( \pi/2 \), have well-known and easily remembered sine and cosine values. Among these special angles is \( \frac{3 \pi}{2} \). The key is to be familiar with the unit circle values:

• \( \cos \frac{3 \pi}{2} = 0 \)
• \( \sin \frac{3 \pi}{2} = -1 \)

Special angles simplify trigonometric calculations, as their sine or cosine values typically become 0, 1, or -1.
These values are particularly useful in trigonometric identities and transformations, as they can often lead to greater simplification, like converting expressions such as \( \cos \left(\frac{3 \pi}{2} + x\right) \) into simpler equations. Understanding and memorizing these special angle values is essential for any trigonometric problem-solving strategy.