Problem 88 - A Graphical Approach to Precalculus with Limits

Question

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(2 \tan ^{-1}(-2)\right)$$

Step-by-Step Solution

Answer

The exact value is \( \frac{-3}{5} \).

Step 1: Understand the Problem

We need to find the exact value of \( \cos \left( 2 \tan^{-1}(-2) \right) \). This involves evaluating an angle that is double the angle whose tangent is \(-2\).

Step 2: Find \( \theta = \tan^{-1}(-2) \)

We identify an angle \( \theta \) such that \( \tan(\theta) = -2 \). This angle occurs in the fourth quadrant, where tangent values are negative.

Step 3: Use Double Angle Formula for Cosine

The formula for the double angle of cosine is \( \cos(2\theta) = 1 - 2\sin^2(\theta) \) or \( \cos(2\theta) = 2\cos^2(\theta) - 1 \). We need \( \sin(\theta) \) and \( \cos(\theta) \) to use these.

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

Since \( \tan(\theta) = \frac{-2}{1} \), we can consider a right triangle with opposite side \(-2\) and adjacent side \(1\). The hypotenuse becomes \( \sqrt{(-2)^2 + 1^2} = \sqrt{5} \). Thus, \( \sin(\theta) = \frac{-2}{\sqrt{5}} \) and \( \cos(\theta) = \frac{1}{\sqrt{5}} \).

Step 5: Calculate \( \cos(2\theta) \)

Using \( \cos(2\theta) = 1 - 2\sin^2(\theta) \), we calculate: \( \sin^2(\theta) = \left(\frac{-2}{\sqrt{5}}\right)^2 = \frac{4}{5} \). Therefore, \( \cos(2\theta) = 1 - 2\left(\frac{4}{5}\right) = 1 - \frac{8}{5} = \frac{-3}{5} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions provide a way to determine an angle given the value of a trigonometric ratio. In our exercise, we encountered an expression involving the inverse tangent function, denoted as \( \tan^{-1} \). This function helps us find the angle \( \theta \) such that the tangent of \( \theta \) equals a specified value, in this case, \(-2\). When we write \( \theta = \tan^{-1}(-2) \), it means we need to find the angle whose tangent value is \(-2\).
In mathematics, the range of the \( \tan^{-1} \) function is restricted to \((-\pi/2, \pi/2)\). This ensures that the angles returned are unique and lie in the first and fourth quadrants. Here, since \( \tan(\theta) = -2\), \(\theta\) must be in the fourth quadrant, where tangent values are negative.

Double Angle Formulas

Double angle formulas are useful trigonometric identities used to find the trigonometric values of double angles. In our exercise, we are tasked with finding \( \cos(2\theta) \) by using the double angle formula for cosine. This formula is expressed as:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \] or equivalently \[ \cos(2\theta) = 1 - 2\sin^2(\theta) \].
Both forms are equally valid and can be used depending on the given trigonometric value information. It simplifies computations by allowing us to use one trigonometric value we know in order to find another. The choice of formula may depend on the available information; however, both require finding \( \sin(\theta) \) and \( \cos(\theta) \) if not already known.

Trigonometric Ratios

Trigonometric ratios are relationships between the sides of a right triangle. In our problem, we encountered the tangent ratio, \( \tan(\theta) \), defined as the ratio of the opposite side to the adjacent side. Given \( \tan(\theta) = -2 \), and using a right triangle, you can depict this ratio as opposite side \(-2\), and adjacent side \(1\). The hypotenuse, calculated using the Pythagorean theorem, is \(\sqrt{5}\).
This allows us to find other trigonometric ratios:

• \( \sin(\theta) = \frac{-2}{\sqrt{5}} \)
• \( \cos(\theta) = \frac{1}{\sqrt{5}} \)

Knowing \( \sin(\theta) \) and \( \cos(\theta) \) aids us in using trigonometric identities to find further trigonometric values necessary for computations.

Exact Trigonometric Values

Exact trigonometric values allow us to calculate precise values without approximations. This is important when interpreting problems involving inverse trigonometric functions or more complex identities. In the problem, our task was to find \( \cos(2\theta) \) using known exact values without a calculator.
First, we determined that \( \sin^2(\theta) = \left( \frac{-2}{\sqrt{5}} \right)^2 = \frac{4}{5} \). Plugging this into the double angle formula \( 1 - 2\sin^2(\theta) \), we find:
\[ \cos(2\theta) = 1 - 2\left(\frac{4}{5}\right) \]
\[ \cos(2\theta) = 1 - \frac{8}{5} = \frac{-3}{5} \]
These exact values help us solve problems algebraically, ensuring accuracy and eliminating rounding errors.