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The inverse sine, also known as \(\text{arcsin}\), is a function that tells us the angle whose sine is a given number. For example, \(\text{arcsin} \left( \frac{\text{\sqrt{3}}}{2} \right)\) gives us the angle \(\frac{\pi}{3}\) because \(\text{sin} \left( \frac{\pi}{3} \right) = \frac{\text{\sqrt{3}}}{2}\). This aspect of the inverse sine is crucial to solve problems where an angle needs to be determined from a sine value. In our exercise, recognizing \(\text{arcsin} \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{3}\) is a key step. It's important to remember:

• \(\text{arcsin}\) is the inverse of sine.
• The range of \(\text{arcsin}\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• \(\text{arcsin} \) converts a sine value back to its corresponding angle.

The cosine function is a fundamental trigonometric function. It relates the ratio of the adjacent side to the hypotenuse in a right triangle to an angle. For an angle \(\theta\), \(\cos(\theta)\) is the length of the adjacent side divided by the hypotenuse. For unit circle and angle calculations, some key cosine values are handy:

• \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\)
• \(\cos(\frac{\pi}{3}) = \frac{1}{2} \)
• \(\cos(0) = 1\)

In our exercise, we need to find the cosine of the angle \(\frac{\pi}{6}\). Knowing that \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\) provides the final answer. Cosine tables or the unit circle method can be used to remember these specific values.

Angle division is a simple but crucial step in many trigonometric problems. In our exercise, we are asked to find:\( \cos \left( \frac{1}{2} \sin^{-1} \left( \frac{\sqrt{3}}{2} \right) \right) \)First, we determine the angle whose sine is \(\frac{\sqrt{3}}{2}\), which is \(\frac{\pi}{3}\). Then, we divide this angle by 2 to get \(\frac{\pi}{6}\). Working with angles and fractions of angles requires:

• Accurate angle values
• Basic fraction skills
• Understanding trigonometric identities

This step helps to simplify and handle complex trigonometric functions easier. Finally, finding \( \cos \left( \frac{\pi}{6} \right) \) completes our problem solution with \( \frac{\sqrt{3}}{2} \). Angle division is key to managing and breaking down trigonometric expressions systematically.