Problem 27 - Precalculus: Functions and Graphs

Question

Write the expression as an algebraic expression in \(x\) for \(x>0\). $$\sin \left(2 \sin ^{-1} x\right)$$

Step-by-Step Solution

Answer

The algebraic expression is \(2x \sqrt{1 - x^2}\).

Step 1: Understand the Function

The given function is \(\sin(2 \sin^{-1} x)\). Here, \(\sin^{-1} x\) is the inverse sine function, also known as arcsin. The expression requires finding \(\sin\) of double the angle given by \(\sin^{-1} x\).

Step 2: Use Double Angle Identity

Recall the double angle identity for sine: \(\sin(2\theta) = 2 \sin \theta \cos \theta\). In our problem, \(\theta = \sin^{-1} x\). Therefore, \(\sin(2\sin^{-1} x) = 2 \sin(\sin^{-1} x) \cos(\sin^{-1} x)\).

Step 3: Simplify \(\sin(\sin^{-1} x)\)

By definition of inverse, \(\sin(\sin^{-1} x) = x\). Therefore, the expression simplifies to \(2x \cos(\sin^{-1} x)\).

Step 4: Find \(\cos(\sin^{-1} x)\)

The relation \(\cos(\theta) = \sqrt{1 - \sin^2(\theta)}\) can be used here. Since \(\theta = \sin^{-1} x\), \(\cos(\sin^{-1} x) = \sqrt{1 - x^2}\). This is valid for \(x > 0\).

Step 5: Combine and Simplify the Expression

Substitute \(\cos(\sin^{-1} x)\) from Step 4 back into the expression: \(2x \cos(\sin^{-1} x) = 2x \sqrt{1 - x^2}\). Thus, the algebraic expression is \(2x \sqrt{1 - x^2}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, also called arc functions, perform the opposite operation to regular trigonometric functions. They help us find the angle whose sine, cosine, or tangent is a given number. For example, if we know that the sine of an angle \(\theta\) is x, then \(\sin^{-1}(x)\), also known as arcsin, will give us \(\theta\).
This concept is crucial when dealing with expressions like \(\sin(2\sin^{-1} x)\). The arcsin function will "undo" the sine and provide the angle. Once we have the angle, we can apply trigonometric identities or other operations. Keep in mind that, due to the limitations of inverse trigonometric functions, \(x\) must be within a certain range (e.g., \(-1 \leq x \leq 1\) for arcsin). This ensures that the resulting angle falls within the principal range of the inverse function. In our problem, since \(x > 0\), we're in a valid range, simplifying our calculations.

• The result of an inverse trigonometric function is an angle.
• The domain must be considered when working with these functions.
• These functions are essential for solving trigonometric equations that involve angles.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In the context of trigonometry, we often need to convert trigonometric expressions into simpler algebraic forms to solve them more easily.
For instance, in the exercise, we transform \(\sin(2\sin^{-1} x)\) into an algebraic expression. This requires identifying equivalent algebraic representations. Initially, we use the identity \(\sin(\sin^{-1} x) = x\). This results directly from the definition of inverse trig functions. Then, we employ the identity \(\cos(\sin^{-1} x) = \sqrt{1 - x^2}\), which uses the Pythagorean identity for sine and cosine.
Basically, converting trigonometric expressions into algebraic terms allows faster calculations, and it often makes the problem easier to understand.

• Algebraic expressions follow basic arithmetic rules.
• They provide an empirical way to manipulate and solve math problems.
• Transitions from trigonometric to algebraic forms often involve identities.

Double Angle Identity

The double angle identity is one of the critical trigonometric identities used frequently when dealing with expressions like \(\sin(2\theta)\). The identity \(\sin(2\theta) = 2 \sin \theta \cos \theta\) is especially useful because it relates a trigonometric function of double the angle to the trigonometric functions of the single angle.
In practical terms, this identity helps break down complex trigonometric expressions into simpler parts. For the given problem, once we establish \(\theta = \sin^{-1} x\), applying the double angle identity is straightforward. It allows us to use known trigonometric values for \(\sin\theta\) and \(\cos\theta\) to find the sine of double that angle.Remember that employing such identities often needs simultaneous application of other identities, such as using the Pythagorean theorem to find the cosine component. The consistent use of these identities makes what seems like a complicated problem feel approachable and manageable.

• Double angle identities are vital for simplifying expressions.
• They reduce complex equations to simpler forms.
• Combining with other identities often provides elegant solutions.