Problem 47 - Algebra and Trigonometry

Question

\(46-47\) . Graphing Inverse Trigonometric Functions (a) Graph the function and make a conjecture, and (b) prove that your conjecture is true. $$ y=\tan ^{-1} x+\tan ^{-1} \frac{1}{x} $$

Step-by-Step Solution

Answer

The function \(y = \tan^{-1} x + \tan^{-1} \frac{1}{x}\) equals \(\frac{\pi}{2}\) for all \(x > 0\), and equals \(-\frac{\pi}{2}\) for \(x < 0\), excluding \(x = 0\).

Step 1: Understanding the Function

We need to analyze the function \(y = \tan^{-1} x + \tan^{-1} \frac{1}{x}\). The inverse tangent function (or arctan) has some specific properties, like \(\tan^{-1} x\) returning angles whose tangent is \(x\). Understanding how \(\tan^{-1}\) responds to its arguments is critical to predicting and verifying behavior graphically and algebraically.

Step 2: Analyze the Behavior of Each Term

Recall that \(\tan^{-1}(x)\) increases from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) as \(x\) goes from negative infinity to positive infinity. Consider how \(\tan^{-1}\left(\frac{1}{x}\right)\) behaves: it flips over at \(x = 1\) with undefined behavior at \(x = 0\). This suggests examining the symmetry and potential simplifications.

Step 3: Graph the Function

Graphing can help visually inspect the function. When you plot \(y = \tan^{-1} x + \tan^{-1} \frac{1}{x}\) (ensuring not to include \(x = 0\) where it is undefined), you observe horizontal symmetry about \(y = \frac{\pi}{2}\). For large \(|x|\), both terms approach integer multiples of \(\frac{\pi}{2}\) resulting in \(y\) approaching \(\frac{\pi}{2}\).

Step 4: Formulate a Conjecture

Based on observation, conjecture that \(\tan^{-1} x + \tan^{-1} \frac{1}{x} = \frac{\pi}{2}\) for all \(x > 0\) or \(\tan^{-1} x + \tan^{-1} \frac{1}{x} = -\frac{\pi}{2}\) for \(x < 0\), excluding \(x = 0\).

Step 5: Prove the Conjecture

Use the identity involving the sum of inverse tangents: \(\tan^{-1} u + \tan^{-1} v = \tan^{-1}\left(\frac{u + v}{1 - uv}\right)\), applicable when \(uv < 1\). Set \(u = x\) and \(v = \frac{1}{x}\): the expression becomes \(\tan^{-1}\left(\frac{x + \frac{1}{x}}{1 - x \cdot \frac{1}{x}}\right) = \tan^{-1}\left(\frac{x + \frac{1}{x}}{0}\right)\), which simplifies directly showing that as long as \(x > 0\), the identity \(y = \frac{\pi}{2}\) holds.

Key Concepts

Arctan Properties

The arctan function, short for arc tangent, is an inverse trigonometric function that helps us find the angle when given the tangent value. It's important because it reverses the tangent function, allowing us to work backwards from the ratio back to the angle.

• Range and Domain: Arctan covers angles from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This means it returns angles in this range, particularly useful in many scenarios, including the ranges of specific quadrants.
• Behavior: As \(x\) approaches positive or negative infinity, arctan \(x\) nears \(\frac{\pi}{2}\) or \(-\frac{\pi}{2}\) respectively. This makes the function approach horizontal asymptotes.
• Properties: One notable property of arctan is shown when calculating the sum of two arctan functions: \(\tan^{-1} u + \tan^{-1} v = \tan^{-1}\left(\frac{u+v}{1-uv}\right)\), applicable when \(uv < 1\). This identity is key in simplifying expressions involving sums of inverse tangents.

Function Graphing

Visualizing the arctan function helps in understanding its behavior and identifying key characteristics. When graphing functions like \(y = \tan^{-1} x + \tan^{-1} \frac{1}{x}\), we gain insights into symmetry and limits.

• Graph Shape: By plotting, the function appears symmetric about \(y = \frac{\pi}{2}\), showing a curved approach towards horizontal asymptotes as \(x\) extends towards infinity.
• Graph Features: It's notably undefined at \(x = 0\) due to division by zero in \(\frac{1}{x}\), creating a gap or discontinuity in the graph. Observing this visually emphasizes the limitation in the domain.
• Limit Observations: As \(x\) increases or decreases significantly, the individual arctan terms adjust towards their respective limits, resulting in the entire function stabilizing at \(\frac{\pi}{2}\) for positive \(x\) and \(-\frac{\pi}{2}\) for negative \(x\).

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the initial variables, linking angles and side lengths of triangles. They're incredibly useful in simplifying complex trigonometric equations and proving conjectures.

• Inverse Identities: When dealing with inverses like \(\tan^{-1} x\), we often need identities to handle equations with multiple terms. For instance, \(\tan^{-1} u + \tan^{-1} v = \tan^{-1}\left(\frac{u+v}{1-uv}\right)\) is pivotal in verifying relationships involving sums of arctan functions.
• Sum and Difference Identities: These involve the sine, cosine, and tangent functions, helping simplify the addition or subtraction of two angles into a single trigonometric expression.
• Verification: Identities aid in proving conjectures by transforming complicated trigonometric components into simpler, digestible expressions, making it easier to validate equations, such as the pairing of arc functions leading to expected results like \(y = \frac{\pi}{2}\).