Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-30: Write the expression as an algebraic expression in \(x\) for \(x>0\). $$ \tan \left(\frac{1}{2} \cos ^{-1} \frac{1}{x}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Goal

The task is to express \( \tan \left(\frac{1}{2} \cos ^{-1} \frac{1}{x}\right) \) as an algebraic function of \( x \) for \( x > 0 \). This involves rewriting the trigonometric expression in terms of algebraic operations involving \( x \).

Step 2: Applying Half-Angle Identities

We can use the half-angle identity for tangent: \( \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta} \). In this case, \( \theta = \cos^{-1} \left(\frac{1}{x}\right) \). This means we need to find expressions for \( \cos\theta \) and \( \sin\theta \).

Step 3: Evaluating \( \cos \theta \)

Since \( \theta = \cos^{-1} \left(\frac{1}{x}\right) \), we have \( \cos \theta = \frac{1}{x} \).

Step 4: Finding \( \sin \theta \) Using Pythagorean Identity

Utilize the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \sin \theta \). This gives \( \sin \theta = \sqrt{1 - \left(\frac{1}{x}\right)^2} = \sqrt{\frac{x^2 - 1}{x^2}} = \frac{\sqrt{x^2-1}}{x} \).

Step 5: Substitute and Simplify

Substitute \( \cos \theta \) and \( \sin \theta \) into the half-angle identity. The result is: \( \tan\left(\frac{\theta}{2}\right) = \frac{1 - \frac{1}{x}}{\frac{\sqrt{x^2-1}}{x}} = \frac{x - 1}{\sqrt{x^2-1}} \). This can be simplified to \( \frac{x - 1}{\sqrt{x^2-1}} \).

Key Concepts

Half-Angle Identities

Half-angle identities are mathematical tools used in trigonometry to simplify expressions involving angles that are half the size of a known angle. They are derived from double-angle identities, which describe the trigonometric functions of twice an angle. The half-angle identity is crucial when you need to calculate the trigonometric function of an angle expressed as half of another angle.

For example, when dealing with tangent, the half-angle identity is given by:

• \(\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta}\)

This identity helps when you have information about \(\cos\theta\) but need to find the tangent of half that angle. By knowing the half-angle identity for tangent, you're equipped to break down complex trigonometric expressions and rewrite them in simpler algebraic terms.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angle that corresponds to a particular trigonometric value. In essence, they reverse what the basic trigonometric functions do. For example, if \( \cos\theta = \, ext{some value} \, \), then \( \theta = \cos^{-1}\left(\text{that value}\right)\).

In the given problem, the inverse cosine function, \( \cos^{-1} \left(\frac{1}{x}\right) \), helps determine the angle whose cosine is \( \frac{1}{x} \). This angle, denoted as \( \theta \), forms the basis for applying further trigonometric identities, like the half-angle identity. Understanding inverse functions is essential, as they provide the angle input necessary for utilizing other trigonometric identities.

Pythagorean Identity

The Pythagorean identity is one of the fundamental properties in trigonometry related to right triangles. It states:

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

This identity helps to find one trigonometric function when another is known. For instance, if you know \( \cos\theta \) and need to find \( \sin\theta \), the Pythagorean identity offers a pathway for solving.

• \( \sin\theta = \sqrt{1 - \cos^2\theta} \)

In our solution, after finding \( \cos\theta = \frac{1}{x} \), the Pythagorean identity allows for determining \( \sin\theta = \frac{\sqrt{x^2-1}}{x} \). This crucial step gives us the components needed for applying the half-angle identity, ensuring the trigonometric expression is rewritten as an algebraic one.

Trigonometric Expressions

Trigonometric expressions involve various functions, such as sine, cosine, and tangent, often within a larger equation or formula. Solving a trigonometric expression can involve rewriting it in terms of algebraic expressions, using identities like half-angle or Pythagorean, as seen in the exercise.

For example, converting \( \tan\left(\frac{1}{2} \cos^{-1} \frac{1}{x}\right) \) to an algebraic expression involves expressing all functions in terms of \( x \), fully simplifying the part of the trigonometric function. These tasks utilize known values like \( \cos\theta \) and identities to restate \( \tan \left( \frac{1}{2} \cos^{-1} \frac{1}{x}\right) \) purely in terms of algebraic operations involving \( x \).

• The transformed expression, \( \frac{x - 1}{\sqrt{x^2-1}} \), combines initial trigonometric insights into a purely algebraic framework. Understanding this transformation is essential for mastering complex trigonometric identities in algebraic forms.