Problem 50 - Precalculus Mathematics for Calculus

Question

Find the exact value of the given expression. $$\tan \left(\frac{1}{2} \cos ^{-1} \frac{2}{3}\right)$$

Step-by-Step Solution

Answer

The exact value is \( \frac{1}{\sqrt{5}} \) or \( \frac{\sqrt{5}}{5} \) after rationalizing.

Step 1: Understand the Problem

We need to find the value of \( \tan \left(\frac{1}{2} \cos^{-1} \frac{2}{3}\right) \). This involves inverse trigonometric functions and the identity for tangent half-angle.

Step 2: Use the Identity for Half Angle Tangent

Recall the identity for tangent of half an angle: \( \tan \left( \frac{\theta}{2} \right) = \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \). In our problem, \( \theta = \cos^{-1} \frac{2}{3} \). We will use this identity to simplify the expression.

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

Since \( \theta = \cos^{-1} \frac{2}{3} \), we have \( \cos \theta = \frac{2}{3} \). This is given directly in the problem.

Step 4: Calculate the Expression Using Identity

Substitute \( \cos \theta = \frac{2}{3} \) into the half-angle identity:\[\tan \left( \frac{1}{2} \cos^{-1} \frac{2}{3} \right) = \sqrt{\frac{1 - \frac{2}{3}}{1 + \frac{2}{3}}} = \sqrt{\frac{1/3}{5/3}}\]This simplifies to:\[\sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}}\]Therefore, \( \tan \left( \frac{1}{2} \cos^{-1} \frac{2}{3} \right) = \frac{1}{\sqrt{5}} \).

Step 5: Rationalize the Denominator (if necessary)

If needed, we can rationalize \( \frac{1}{\sqrt{5}} \) to make it \( \frac{\sqrt{5}}{5} \). However, depending on the context or instructions given in a problem, the answer might be left in either form.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are unique functions that allow us to determine the angle measurement from a given trigonometric value. These functions are pivotal when dealing with inverse calculations. In this exercise, we focus on \( \cos^{-1} \frac{2}{3} \), which finds the angle \( \theta \) such that the cosine of that angle is \( \frac{2}{3} \).
The notation \( \cos^{-1} \) represents the arccosine, commonly used to switch from a cosine value back to an angle. It is important to remember that inverse trigonometric functions will return an angle within a specific range. For cosine, this range is [0, π], meaning any angle produced will reside within these boundaries.
Understanding this function provides the foundation to then apply further trigonometric identities and simplifications. Therefore, it's crucial to become familiar with converting between functions and angles using inverse operations.

Rationalizing Denominators

Rationalizing a denominator means transforming a fraction so that its denominator contains no irrational numbers, like radicals. This process sometimes becomes necessary in order to simplify expressions into more usable forms.
In the solution provided, after simplifying \( \sqrt{\frac{1}{5}} \) to \( \frac{1}{\sqrt{5}} \), it might be rationalized to \( \frac{\sqrt{5}}{5} \). This is done by multiplying the numerator and the denominator by \( \sqrt{5} \).

• Multiply: \( \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} \)
• Now the denominator is rationalized since it no longer includes a radical.

Rationalizing is frequently used in mathematics to ensure that answers are presented in the simplest form possible. This helps maintain clarity in calculations and in the formality of mathematical solutions.

Trigonometric Identities

Trigonometric identities are equations that connect the trigonometric functions. They provide the basis for calculating various trigonometric values and simplifying complex expressions.
In this exercise, we utilized the half-angle identity for tangent \( \tan \left( \frac{\theta}{2} \right) = \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \). This identity helps simplify the expression involving an inverse cosine function into a straightforward evaluation.

• First, by substituting \( \cos \theta = \frac{2}{3} \) from the given value.
• The expression becomes: \( \tan \left( \frac{1}{2} \cos^{-1} \frac{2}{3} \right) = \sqrt{\frac{1 - \frac{2}{3}}{1 + \frac{2}{3}}} \).
• Upon simplifying: \( \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \).

These identities are essential for solving trigonometry problems efficiently and are a necessary tool in any student's mathematical toolbox. Learning and practicing these can greatly enhance numerical flexibility and problem-solving skills.