91Ó°ÊÓ

An algebraic expression represents numbers and operations using variables and constants. In this exercise, our goal was to rewrite the trigonometric expression \( \sin(2 \sin^{-1} x) \) in terms of an algebraic expression solely in \( x \). This means transforming the trigonometric expression into an equation that only involves \( x \) and not any trigonometric functions.

An algebraic expression can involve:

• Numerical constants like \(2\)
• Variables like \(x\)
• Operations such as addition, subtraction, multiplication, and division
• Radicals (square roots)

For this problem, through the application of trigonometric identities and properties, we derived that \( \sin(2 \sin^{-1} x) = 2x\sqrt{1-x^2} \), which is a pure algebraic expression in the variable \( x \). This result is particularly useful because algebraic expressions are often simpler to work with than trigonometric forms.

The double angle identity is a crucial trigonometric identity used to express functions of double angles in simpler forms. The identity for sine, specifically, states:

• \( \sin(2\theta) = 2\sin\theta\cos\theta \).

This identity lets us express \( \sin(2\theta) \) in terms of the product of \( \sin\theta \) and \( \cos\theta \), doubling the angle \( \theta \) within the sine function.

In our exercise, once we established \( \theta = \sin^{-1} x \) and thus \( 2\theta = 2\sin^{-1} x \), we could apply this double angle identity to simplify the expression \( \sin(2\theta) \). This allowed us to transform the problem from dealing with a trigonometric inverse and a double angle to a more straightforward multiplication and simplification task within algebraic terms.

The Pythagorean identity is a foundation stone in trigonometry and relates the sine and cosine of a given angle. This identity states:

• \( \sin^2\theta + \cos^2\theta = 1 \).

This identity is vital for converting between trigonometric functions of the same angle.

In the given problem, after identifying that \( \sin\theta = x \), the Pythagorean identity allowed us to find an expression for \( \cos\theta \). Rearranging the identity, we obtained:

• \( \cos^2\theta = 1 - \sin^2\theta \)
• Thus, \( \cos \theta = \sqrt{1 - x^2} \), assuming \(x > 0\).

This insight was instrumental in deriving the final algebraic expression, allowing us to replace trigonometric expressions with their algebraic counterparts effectively.