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Chapter 5: Problem 80

Explain why \(\tan \pi=0\) does not imply that \(\arctan 0=\pi\).

Short Answer

Expert verified
The statement \(\arctan 0=\pi\) is untrue because the range of the \(\arctan\) function is \(-\pi/2 < x < \pi/2\), so \(\arctan\) of any number can not be \(\pi\).

Step by step solution

01

Understanding the \(\tan\) function

As we know, \(\tan(x) = \sin(x) / \cos(x)\). For \(x = \pi\), since \(\sin(\pi) = 0\) and \(\cos(\pi) = -1\), then \(\tan(\pi) = 0 / -1 = 0.\) Therefore, \(\tan \pi = 0\).
02

Understanding the \(\arctan\) function

\(\arctan\) is the inverse of the \(\tan\), which means that \(\arctan(\tan(x)) = x\), but this is only true for \(x\) in the interval \(-\pi/2 < x < \pi/2\). This is because the range of \(\arctan\) function is (-\(\pi/2, \pi/2)\). Since \(\pi\) is not in this interval, \(\arctan 0 = \pi\) can never be true.
03

Why \(\arctan 0 \neq \pi\)

Since 0 lies in the defined range for \(\arctan\), i.e., (-\(\pi/2, \pi/2)\), the result of \(\arctan 0\) would be an angle in this range whose tangent gives 0. From the unit circle, we know that this angle is 0, not \(\pi\). Therefore, \(\arctan 0 = 0\), not \(\pi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function
The tangent function, often represented as \( \tan(x) \), is a fundamental trigonometric function that relates an angle \( x \) in a right-angled triangle to the ratios of two sides. Specifically, it is the ratio of the opposite side to the adjacent side. This function is periodic, meaning it repeats values over a certain range, with a period of \( \pi \), and it is undefined whenever the cosine is zero, as division by zero is not possible.

In the context of the unit circle, which is a circle with a radius one centered at the origin of a coordinate plane, the tangent function projects the angle \( x \) from the origin onto a vertical line that intersects the unit circle at a right angle. The length of the segment on this line between the intersection point and the x-axis represents the tangent of \( x \). Tangent鈥檚 unique properties are essential in understanding many aspects of calculus, physics, and engineering. For instance, knowing that \( \tan(x) \) is zero when \( x \) is \( 0 \) or any multiple of \( \pi \) can help solve equations and analyze wave functions.

When visualizing the tangent function on a graph, it shows a pattern of continuous steep curves that approach but never touch the vertical lines known as asymptotes, which occur at the angles where the cosine is zero and the tangent is undefined, such as \( \frac{\pi}{2} \) and its odd multiples.
Inverse Trigonometric Functions
Inverse trigonometric functions, often indicated with the prefix 'arc' such as \( \arctan \), \( \arcsin \), and \( \arccos \), provide angles when the values of the trigonometric functions are given. For example, \( \arctan(y) \) yields the angle whose tangent is \( y \).

Since trigonometric functions are not one-to-one, their inverses can only exist if their domains are restricted. For \( \tan(x) \), the restriction is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), excluding the endpoints. This domain constraint is necessary because for angles outside this range, multiple angles can have the same tangent value.

An important concept to grasp with inverse trigonometric functions is that they return the

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Most popular questions from this chapter

Find the indefinite integral using the formulas of Theorem \(5.20 .\) $$ \int \frac{1}{1-4 x-2 x^{2}} d x $$

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