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Chapter 5: Problem 2

Find the exact value of the given expression in radians. \(\tan ^{-1}(-1)\)

Short Answer

Expert verified
The exact value is \( -\frac{\pi}{4} \).

Step by step solution

01

Understanding the Problem

We need to find the angle \( \theta \) whose tangent is \( -1 \). This means we are looking for the value of \( \theta \) such that \( \tan( \theta ) = -1 \).
02

Using Inverse Tangent Function

The inverse tangent function \( \tan^{-1}(x) \) provides the angle whose tangent is \( x \). Thus, we need \( \tan(\theta) = -1 \) in terms of radians. Recall that \( \tan \) is negative in the second and fourth quadrants.
03

Identifying the Correct Quadrant

In the second quadrant, the angle corresponding to \( \tan(\theta) = -1 \) is \( \theta = \frac{3\pi}{4} \). In the fourth quadrant, the angle is \( \theta = -\frac{\pi}{4} \). Since the principal value for inverse tangent resides in the range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), we choose \( \theta = -\frac{\pi}{4} \).
04

Conclusion

Therefore, \( \tan^{-1}(-1) = -\frac{\pi}{4} \). This is the principal value that satisfies the given equation.

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Key Concepts

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

trigonometric functions
Trigonometric functions are foundational in understanding angles and their relationships in different contexts like geometry and physics. They include sine, cosine, and tangent, among others. Each of these functions connects an angle in a right triangle to ratios of two sides of the triangle. For example, the tangent function (tan) is defined as the ratio of the opposite side to the adjacent side.

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Most popular questions from this chapter

Find the period of the given function. Graph one cycle using Gnuplot. \(y=\sin \frac{x}{3}+2 \cos \frac{3 x}{4}\)

Draw the graph of the given function for \(0 \leq x \leq 2 \pi\). \(y=2-\cos x\)

Show that \(\tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{2}{9}=\tan ^{-1} \frac{1}{2}\).

Find the amplitude, period, and phase shift of the given function. Then graph one cycle of the function, either by hand or by using Gnuplot (see Appendix B). \(y=2 \csc (2 x-1)\)

Use Gnuplot to graph the function \(y=\frac{\sin x}{x}\) from \(x=-4 \pi\) to \(x=4 \pi\). What happens at \(x=0\) ?
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