Problem 44 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(\tan(2\cos^{-1}(x)) = \frac{2x\sqrt{1-x^2}}{2x^2-1}\)

Step 1: Recognize the Identity

The given expression is \( \tan(2 \cos^{-1}(x)) \). We know that \( \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)} \). Here, let's set \( \theta = \cos^{-1}(x) \).

Step 2: Express \( \tan(\theta) \) in terms of \( x \)

Since \( \theta = \cos^{-1}(x) \), we have \( \cos(\theta) = x \). Using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), we find \( \sin^2(\theta) = 1 - x^2 \). Thus, \( \sin(\theta) = \sqrt{1-x^2} \). Therefore, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\sqrt{1-x^2}}{x} \).

Step 3: Substitute into \( \tan(2\theta) \) formula

Now substitute \( \tan(\theta) = \frac{\sqrt{1-x^2}}{x} \) into the formula \( \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)} \).

Step 4: Simplify the Expression

First, calculate \( \tan^2(\theta) = \left(\frac{\sqrt{1-x^2}}{x}\right)^2 = \frac{1-x^2}{x^2} \). Then, compute the denominator: \( 1 - \tan^2(\theta) = 1 - \frac{1-x^2}{x^2} = \frac{x^2 - 1 + x^2}{x^2} = \frac{2x^2-1}{x^2} \).

Step 5: Final Expression for \( \tan(2\theta) \)

Substitute all this back into the formula for \( \tan(2\theta) \): \[ \tan(2\theta) = \frac{2\left(\frac{\sqrt{1-x^2}}{x}\right)}{\frac{2x^2 - 1}{x^2}} \]. Simplify the expression: This becomes \( \frac{2x \sqrt{1-x^2}}{2x^2 - 1} \). Therefore, \( \tan(2 \cos^{-1}(x)) = \frac{2x \sqrt{1-x^2}}{2x^2 - 1} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics. They help us find the angles corresponding to given trigonometric ratios. These functions can be particularly tricky, so let's break them down.
The inverse cosine function, written as \( \cos^{-1}(x) \), is used to determine the angle \( \theta \) whose cosine is \( x \).
This function tells us the angle in a right triangle where the adjacent side to the angle over the hypotenuse equals \( x \).
Consider this example:

• We have \( \theta = \cos^{-1}(x) \), meaning \( \cos(\theta) = x \).
  
• From here, to find \( \tan(\theta) \), we need to use trigonometric identities, which derive from other core concepts.
  

Generally, inverse trigonometric functions are a bridge from ratio back to angle. You can think of them as asking, "What angle gives me this trigonometric value?"

Algebraic Expressions

Algebraic expressions are combinations of numbers and variables using mathematical operations. In our exercise, we start with a trigonometric expression and need to convert it into a pure algebraic form in terms of \(x\).
For instance:

• \( \tan(\theta) \) is initially expressed using trigonometric identities and then rewritten in algebraic terms.
  
• In our example, \( \tan(\theta) = \frac{\sqrt{1-x^2}}{x} \), which is entirely in terms of \(x\).
  

This transformation requires a good grasp of various identities and algebraic manipulation. By rearranging and substituting these identities, people can simplify and express complex expressions.
Algebra allows us to handle such manipulations, providing a clearer view of what is happening mathematically. When working with algebraic expressions, keep track of each transformation step to avoid confusion.

Trigonometric Functions

Trigonometric functions are the core of many mathematical concepts involving angles. These functions—sine, cosine, and tangent—relate angles to side lengths in right triangles and are widely used in various applications.
One important identity used in the exercise is:

• The double-angle formula for tangent: \( \tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)} \).
  
• This formula allows transformation from \( \tan(\theta) \) to \( \tan(2\theta) \) by altering the angle while keeping the relationship coherent.
  

Understanding these functions is crucial because they enable us to connect different mathematical aspects, like moving from geometric interpretations to algebraic forms.
By mastering trigonometric identities and how to manipulate them, we gain the power to simplify expressions, solve equations, and even model real-world scenarios.