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Magazine Chapter 5: Problem 47
(a) Graph the function and make a conjecture, and (b) prove that your conjecture is true. $$y=\tan ^{-1} x+\tan ^{-1} \frac{1}{x}$$
Short Answer
Expert verified
The function \(y = \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right)\) equals \(\frac{\pi}{2}\) for all \(x \neq 0\).
Step by step solution
01
Graph the Function
To graph the function, plot \( y = \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) \). Start by choosing a range of \( x \) values from negative to positive, including fractions and integers, such as \( -10, -1, -0.5, 0.5, 1, 10 \). Compute \( y \) for these values and plot the points on a graph. Observe the shape and behaviour of the graph, especially noting any symmetry, trends, or boundaries. Based on the behavior of the curve, you might conjecture that \( y = \frac{\pi}{2} \) for all \( x eq 0 \).
02
Analyze the Function
Consider the properties of the arctangent function. Recall that \( \tan^{-1}(x) \) and \( \tan^{-1}\left(\frac{1}{x}\right) \) are undefined at \( x = 0 \), but everywhere else they have defined values. Use the identity: \( \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) = \frac{\pi}{2} \), which holds true for all \( x > 0 \) due to the symmetry and properties of the inverse tangent function. The sum of angles whose tangents are reciprocals is \( \frac{\pi}{2} \). For \( x < 0 \), the identity still holds under the conventional extension of inverse tangent values adding to zero.
03
Proof of Conjecture for x > 0
Let \( x > 0 \). Using the identity \( y = \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) \), observe that for two angles \( A = \tan^{-1}(x) \) and \( B = \tan^{-1}\left(\frac{1}{x}\right) \), \( \tan(A) \cdot \tan(B) = 1 \). So, the tangent of their sum is undefined, implying that the sum is \( \frac{\pi}{2} \). Therefore, \( y = \frac{\pi}{2} \). Thus, the conjecture is proven for \( x > 0 \).
04
Proof of Conjecture for x < 0
Let \( x < 0 \). Apply similar logic: for angles \( A = \tan^{-1}(x) \) and \( B = \tan^{-1}\left(\frac{1}{x}\right) \), \( \tan(A + B) \) is undefined, suggesting \( A + B = \frac{\pi}{2} \, (-\frac{\pi}{2}) \), ensuring their sum aligns with angle conventions. Hence, the sum is \( \frac{\pi}{2} \) by definition extension, reinforcing the conjecture for negative values of \( x \).
05
Conclusion
The observed graph suggests that the function \( y = \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) \) equals \( \frac{\pi}{2} \) for all \( x eq 0 \). The conjecture is proven true by using inverse tangent identities and properties, concluding that this result holds for all specified \( x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arctangent Identity
The arctangent function, denoted as \( \tan^{-1}(x) \), is the inverse of the tangent function. It takes a ratio and returns the angle whose tangent is that ratio. A useful identity involving arctangents states: \( \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) = \frac{\pi}{2} \) for all \( x eq 0 \). This identity stems from the property of angles in a right triangle, where the tangent of one angle is the reciprocal of the other, implying their sum is \( \frac{\pi}{2} \).
In essence, the identity reveals a special relationship between two angles that complements each other to sum up to \( \frac{\pi}{2} \). This fundamental property is pivotal for solving problems involving trigonometric equations and underscores the beauty of trigonometric identities in simplifying complex expressions.
In essence, the identity reveals a special relationship between two angles that complements each other to sum up to \( \frac{\pi}{2} \). This fundamental property is pivotal for solving problems involving trigonometric equations and underscores the beauty of trigonometric identities in simplifying complex expressions.
Graphical Representation
Visualizing functions like \( y = \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) \) helps to understand their behavior and properties. By plotting this function, you observe its behavior over different intervals of \( x \). For values \( x > 0 \) and \( x < 0 \), the function consistently approaches \( \frac{\pi}{2} \).
This graphical representation is valuable because it illustrates:
This graphical representation is valuable because it illustrates:
- The asymptotic behavior of the function as it nears undefined points, such as \( x = 0 \).
- Any symmetry apparent in the graph, which simplifies further analysis or proofs.
- The convergence of the function toward a particular value, helping us form conjectures.
Function Symmetry
Function symmetry is a concept where a function mirrors itself across an axis or a point. In the case of the function \( y = \tan^{-1}(x) + \tan^{-1}\left(\frac{1}{x}\right) \), symmetry plays a crucial role.
When observed, the function shows symmetry in the sense that for equivalent positive and negative values of \( x \), the function output remains consistent at \( \frac{\pi}{2} \). This is significant because:
When observed, the function shows symmetry in the sense that for equivalent positive and negative values of \( x \), the function output remains consistent at \( \frac{\pi}{2} \). This is significant because:
- It reveals intrinsic properties of the function, which makes the analysis concise and elegant.
- Symmetry is often used in creating conjectures, as it provides a predictable pattern in otherwise complex problems.
- It aids in the proof of identities by validating that no matter the substitution of \( x \), the result holds true.
