91Ó°ÊÓ

The inverse sine function, often written as \(\sin^{-1}(x)\) or arcsin, is a crucial concept in trigonometry. This function allows us to determine the angle whose sine is a given number. Specifically, when we evaluate \(\sin^{-1}(1/2)\), we are searching for an angle \(\theta\) such that \(\sin(\theta) = 1/2\).
This angle is commonly known from trigonometric tables as \(\pi/6\) in radians or 30 degrees in the unit circle. The inverse sine function only provides angles within the range \([-\pi/2, \pi/2]\), ensuring each angle is unique for a given sine value. Therefore, when asked to determine \(2 \sin^{-1}(1/2)\), you would find \(\theta = \pi/6\) and then multiply by 2 to get \(\pi/3\).
This transformation allows us to utilize this angle in further trigonometric calculations, such as applying the double angle formulas.

The double angle formula is a powerful tool in trigonometry, allowing the expression of trigonometric functions of double angles in terms of functions of single angles. The cosine double angle formula is given by:
\[ \cos(2A) = 1 - 2\sin^2(A) \]
This equation comes in handy when dealing with expressions involving angles that are multiples of others. In our problem, we've determined \(A = \pi/3\).
Using the double angle formula, we analyze the expression \(\cos(2 \times \pi/3)\). First, substitute \(\pi/3\) into the formula:
\[ \cos(2 \times \pi/3) = 1 - 2\sin^2(\pi/3) \]
By doing this, complex expressions become simpler to solve and understand, as we're transforming the problem using known identities. This manipulation is a strategic approach in many trigonometric problems and exercises.

Trigonometric identities are equations involving trigonometric functions that hold true for every value of the variables involved.
These identities help simplify the calculation of trigonometric problems. One well-known identity used here is \(\sin(\pi/3) = \sqrt{3}/2\).

• This value allows us to substitute directly into the double angle formula.
• Once substituted, further simplifications can be made, as seen where \(1 - 2(\sqrt{3}/2)^2\) simplifies to \(1 - 3/2\).
• The final simplification gives us \(-0.5\), the exact value of the original expression.

Understanding and using these identities lead to accurate evaluations of trigonometric expressions. They form the foundation of much more complex trigonometric reasoning and solution strategies.