Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each value without using a calculator $$ \tan \left[2 \tan ^{-1}\left(\frac{1}{3}\right)\right] $$

Step-by-Step Solution

Answer

The value is \(\frac{3}{4}\).

Step 1: Understand the Problem

We need to find the value of \( \tan \left[2 \tan^{-1} \left( \frac{1}{3} \right) \right] \). This involves using the double angle identity for tangent, as the inner function is an inverse tangent.

Step 2: Recall the Tangent Double Angle Formula

The double angle formula for tangent is \( \tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)} \). We will use this formula with \( \theta = \tan^{-1} \left( \frac{1}{3} \right) \).

Step 3: Set \( \theta = \tan^{-1} \left( \frac{1}{3} \right) \)

If \( \theta = \tan^{-1} \left( \frac{1}{3} \right) \), then \( \tan(\theta) = \frac{1}{3} \). Substitute this value into the double angle formula.

Step 4: Substitute into Tangent Double Angle Formula

Substituting the known value, \( \tan(\theta) = \frac{1}{3} \), into the double angle formula:\[ \tan(2\theta) = \frac{2 \cdot \frac{1}{3}}{1 - \left(\frac{1}{3}\right)^2} = \frac{\frac{2}{3}}{1 - \frac{1}{9}} \]

Step 5: Simplify the Expression

First, simplify the denominator: \(1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9}\).Now perform the division in the fraction:\[ \frac{\frac{2}{3}}{\frac{8}{9}} = \frac{2}{3} \times \frac{9}{8} = \frac{2 \times 9}{3 \times 8} = \frac{18}{24} = \frac{3}{4} \]

Step 6: Write the Final Answer

Thus, the value of \( \tan \left[2 \tan^{-1} \left( \frac{1}{3} \right) \right] \) is \( \frac{3}{4} \).

Key Concepts

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions are essential in solving many geometric and trigonometric problems. They help us determine the angle when given a ratio from a trigonometric function. Here, when we see \( \tan^{-1} \left( \frac{1}{3} \right) \), this refers to the angle \( \theta \) such that the tangent of \( \theta \) is \( \frac{1}{3} \).

These functions essentially "undo" what the normal trigonometric functions do:

• \( \tan^{-1}(x) \) gives the angle whose tangent is \( x \).
• \( \sin^{-1}(x) \) finds the angle whose sine is \( x \).
• \( \cos^{-1}(x) \) calculates the angle whose cosine is \( x \).

For each inverse trigonometric function, we define a specific range in which the function provides values, ensuring that each input corresponds to a unique output. This is crucial because trigonometric functions are periodic, meaning they repeat their values over certain intervals. Therefore, the inverse functions are only defined for principal values, helping us avoid ambiguities.

Exploring Key Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable where both sides of the equation are defined. They are essential tools for simplifying expressions and solving trigonometric equations. In our exercise, we utilize the tangent double angle identity.

The tangent double angle formula is:\[\tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)}\]This identity helps us relate the tangent of a double angle to the tangent of the original angle. The formula breaks down the complex angle \( 2\theta \) into simpler components based on \( \tan(\theta) \).

Other important identities include:

• Pythagorean identities, like \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Sum and difference formulas, which allow calculations for angles like \( \theta + \phi \).
• Reciprocal identities such as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

Each of these identities helps us solve various types of trigonometric problems by transforming more complex expressions into simpler or more familiar forms.

Mastering Angle Simplification Methods

Angle simplification methods are techniques used to simplify trigonometric expressions involving angles. These methods often involve breaking down complex expressions by separating or combining angles in a way that utilizes trigonometric identities.

In the exercise given, we simplify the angle by using the tangent double angle identity, which efficiently computes the tangent of double the given angle. It involves understanding how smaller parts of the angle can be reassembled using the identities.

Key strategies for angle simplification include:

• Converting the expression using known identities, as shown with \( \tan(2\theta) \).
• Breaking down complicated angles into sums or differences of known angles.
• Utilizing properties of symmetry in trigonometric functions, such as the fact that \( \sin(-\theta) = -\sin(\theta) \).

These simplification techniques are essential for making trigonometric calculations manageable without calculators, especially in solving higher-level problems.