Problem 46 - Algebra and Trigonometry

Question

Rewrite the expression as an algebraic expression in \(x .\) \(\sin \left(2 \sin ^{-1} x\right)\)

Step-by-Step Solution

Answer

\( \sin (2 \sin^{-1} x) = 2x\sqrt{1-x^2} \).

Step 1: Identify the Inner Function

Recognize that the inner function is \( \sin^{-1} x \), which is the inverse of the sine function. This function returns an angle \( \theta \) such that \( \sin(\theta) = x \). Therefore, let \( \theta = \sin^{-1} x \), which implies that \( \sin(\theta) = x \).

Step 2: Consider the Outer Function

The outer function is \( \sin(2\theta) \). Use a trigonometric identity to express \( \sin(2\theta) \) in terms of \( \sin(\theta) \) and \( \cos(\theta) \). The identity is \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \).

Step 3: Express \(\cos(\theta)\) in Terms of \(x\)

Since \( \theta = \sin^{-1} x \) and \( \sin(\theta) = x \), we know from the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) that \( \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - x^2 \). Therefore, \( \cos(\theta) = \sqrt{1-x^2} \) assuming \( \theta \) is in the principal range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), where \( \cos(\theta) \geq 0 \).

Step 4: Substitute Back into the Expression

Substitute \( \sin(\theta) = x \) and \( \cos(\theta) = \sqrt{1-x^2} \) back into the identity from Step 2: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) = 2x\sqrt{1-x^2} \).

Step 5: Final Expression

Thus, the expression \( \sin \left(2 \sin^{-1} x\right) \) simplifies to \( 2x\sqrt{1-x^2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when the values of trigonometric functions like sine, cosine, or tangent are known. They essentially reverse the action of the standard trigonometric functions. For instance:

• The function \( \sin^{-1}(x) \) is the inverse sine function, usually referred to as arcsine. It gives us the angle \( \theta \) whose sine is \( x \).
• If \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \).

Understanding these inverse functions is vital for solving trigonometric equations and rewriting expressions in algebraic terms. When dealing with inverse trigonometric functions, it’s important to remember their range of outputs. For \( \sin^{-1}(x) \), the principal range is \([-\pi/2, \pi/2]\), which ensures that the output angle \( \theta \) is restricted such that \( \cos(\theta) \) remains non-negative.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for all angles. They help simplify expressions and solve equations. One of the most useful identities, especially when dealing with double angles, is:

• The double angle identity for sine: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \).

This identity provides a way to express \( \sin(2\theta) \) using \( \sin(\theta) \) and \( \cos(\theta) \). This is particularly useful when given an expression like \( \sin(2\sin^{-1}(x)) \), because it allows us to substitute values of \( \sin(\theta) \) and \( \cos(\theta) \) to find an algebraic representation. By using this identity effectively, complex trigonometric expressions can be reduced to simpler, more workable forms, often in terms of a single variable.

Pythagorean Identity

The Pythagorean identity is one of the fundamental tools in trigonometry. It connects the squares of the sine and cosine of an angle, giving us:

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

This identity is derived from the Pythagorean theorem and holds true for any angle \( \theta \). In terms of rewriting trigonometric expressions, this identity allows us to express \( \cos(\theta) \) in terms of \( \sin(\theta) \).
For example, if \( \sin(\theta) = x \), then \( \cos^2(\theta) = 1 - x^2 \), resulting in \( \cos(\theta) = \sqrt{1 - x^2} \).
This relationship makes it possible to transform trigonometric expressions into algebraic forms that include square roots and squares, as seen in expressions like \( 2x\sqrt{1-x^2} \), which appear when applying the double angle identity or solving problems involving inverse trigonometric functions.