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Chapter 6: Problem 74

Evaluate each expression exactly, if possible. If not possible, state why. $$\tan ^{-1}[\tan (8 \pi)]$$

Short Answer

Expert verified
The expression evaluates to 0.

Step by step solution

01

Understand the expression

The exercise asks us to evaluate the expression \(\tan^{-1}[\tan(8\pi)]\). This involves understanding how the \(\tan\) and \(\tan^{-1}\) functions interact.
02

Evaluate \(\tan(8\pi)\)

The tangent function \(\tan(\theta)\) has a period of \(\pi\). Therefore, \(\tan(8\pi) = \tan(0) = 0\) because \(8\pi\) is equivalent to 0 radian over complete cycles.
03

Apply the inverse tangent function

The inverse tangent function \(\tan^{-1}(x)\) gives the angle whose tangent is \(x\). We have \(\tan^{-1}(0)\). Tangent of angle 0 is 0, so \(\tan^{-1}(0) = 0\).
04

Conclusion

The expression \(\tan^{-1}[\tan(8\pi)]\) simplifies to \(0\), because \(\tan^{-1}(0) = 0\).

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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 denoted as \( \tan(\theta) \), is a fundamental trigonometric function. It is defined as the ratio of the sine and cosine functions: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This function relates an angle of a right triangle to the ratio of the length of the side opposite the angle to the side adjacent.
  • Range: The range of the tangent function spans from \(-\infty\) to \(+\infty\), or all real numbers. This is because the values of the ratio \( \frac{\sin(\theta)}{\cos(\theta)} \) can vary widely as \( \theta \) changes.
  • Behavior: Importantly, the tangent function is undefined when \( \cos(\theta) = 0 \), because division by zero in mathematics is undefined. Thus, tangent has vertical asymptotes where the angle corresponds to \( 90^\circ + k\times180^\circ \), where \( k \) is an integer.
Understanding the tangent function involves recognizing these properties and appreciating how they influence the function's graphical representation and application in solving problems.
Inverse Tangent
The inverse tangent, also denoted as \( \tan^{-1}(x) \) or \( \arctan(x) \), is the opposite of the tangent function. This function takes a real number \( x \) as its input and returns an angle \( \theta \) as its output, satisfying \( \tan(\theta) = x \).
  • Range: The output, or range, of the inverse tangent function is restricted to \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), or \(-90^\circ\) to \(90^\circ\). This restriction ensures that each output angle is unique for a corresponding \( x \).
  • Usage: It is mainly used to determine an angle when the tangent value is known. For example, if \( \tan(\theta) = x \), then \( \theta = \tan^{-1}(x) \) gives the precise measure of the angle.
The inverse tangent function helps bridge the gap between the ratio provided by the tangent and the angle it represents, an essential concept in solving trigonometric equations.
Periodicity of Tangent Function
One of the noteworthy properties of the tangent function is its periodicity. A function is periodic if it repeats its values in regular intervals or periods. The tangent function has a period of \( \pi \), which means it repeats its values every \( \pi \) radians.
  • Implication: If you have \( \tan(\theta) \), then \( \tan(\theta + n\pi) = \tan(\theta) \) for any integer \( n \). This periodicity is due to the nature of the sine and cosine functions, which make up the tangent function and also show periodic behavior.
  • Example in Problem: In the given problem, we evaluated \( \tan(8\pi) \). Using periodicity, we realize that this is equivalent to \( \tan(0) \), given \( 8\pi \) is a multiple of \( \pi \). Therefore, the function value returns to that of \( \tan(0) \), which is 0.
The predictability of periodicity simplifies the calculation of tangent values for angles beyond the basic cycle of \( 0 \) to \( \pi \). It is a practical tool for trigonometric evaluations.

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