Problem 43 - Algebra and Trigonometry

Question

\(43-46\). Write the given expression as an algebraic expression in \(x\). $$ \sin \left(2 \tan ^{-1} x\right) $$

Step-by-Step Solution

Answer

\( \sin(2 \tan^{-1} x) = \frac{2x}{x^2 + 1} \)

Step 1: Understanding the Expression

The given expression is \( \sin(2 \tan^{-1} x) \). This involves applying a double angle identity for sine, where the angle is expressed with an inverse tangent function.

Step 2: Use Trigonometric Identities

Recall the double angle identity for sine: \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \). We will apply this identity by treating \( \theta = \tan^{-1} x \).

Step 3: Express \( \sin(\theta) \) and \( \cos(\theta) \) with \( x \)

Since \( \theta = \tan^{-1} x \), we have \( \tan(\theta) = x \). If we consider a right triangle, \( \tan(\theta) = x = \frac{x}{1} \), so opposite = \(x\) and adjacent = \(1\). The hypotenuse will be \( \sqrt{x^2 + 1} \).

Step 4: Calculate \( \sin(\theta) \) and \( \cos(\theta) \)

From the triangle, \( \sin(\theta) = \frac{x}{\sqrt{x^2 + 1}} \) and \( \cos(\theta) = \frac{1}{\sqrt{x^2 + 1}} \).

Step 5: Substitute into Double Angle Formula

Using the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \), we have:\[ \sin(2 \tan^{-1} x) = 2 \times \frac{x}{\sqrt{x^2 + 1}} \times \frac{1}{\sqrt{x^2 + 1}} \]

Step 6: Simplify the Expression

Combine the fractions:\[ = 2 \times \frac{x}{x^2 + 1} = \frac{2x}{x^2 + 1} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools used in mathematics to relate various trigonometric functions to one another. They are equations that hold true for all values of the involved variables. Some common identities include the Pythagorean identities, sum and difference formulas, and double angle formulas. Each identity helps simplify complex trigonometric expressions or solve trigonometric equations.

In the given exercise, we focus on the double angle identity for sine, one of the trigonometric identities:

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

This identity allows us to express the sine of double the angle \(\theta\) in terms of sine and cosine of \(\theta\). Understanding and applying such identities is crucial in transforming trigonometric expressions into a more manageable form. This practice is particularly useful in calculus, physics, and engineering.

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse functions of trigonometric functions and are crucial in solving trigonometric equations. They allow us to determine the angle for a given trigonometric value. For example, \( \tan^{-1}(x) \) or arctangent of \(x\) gives us an angle whose tangent is \(x\). These functions can be particularly handy when working with angles that are derived from coordinate-based situations.

In this exercise, \( \theta = \tan^{-1}(x) \) is used, and this indicates an angle whose tangent value is \(x\).

• Useful in various applications, including geometry and calculus
• Often represented graphically as arcs

Understanding inverse trigonometric functions enables students to work seamlessly between trigonometric ratios and angles, thereby simplifying complex transformations of trigonometrically defined situations.

Double Angle Formulas

One important set of identities in trigonometry is the double angle formulas. These are used to express trigonometric functions of double angles, i.e., twice a particular angle. They make it possible to break down more complex expressions into simpler ones. The formula for doubling the angle of the sine function is:

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

This identity is particularly useful when one needs to express trigonometric functions involving double angles in terms of single angles.

In our given expression \( \sin(2 \tan^{-1} x) \), the double angle formula allows us to reformulate it into something more straightforward to compute. Instead of dealing directly with a double angle that involves an inverse trigonometric function, we express it using simpler sine and cosine functions.
This simplification helps in computations and provides a deeper insight into the transformation properties of trigonometric functions.