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Chapter 1: Problem 55

Multiple Choice Which of the following is the measure of \(\tan ^{-1}(-\sqrt{3})\) in degrees? \(\begin{array}{llll}{\text { (A) }-60^{\circ}} & {\text { (B) }-30^{\circ}} & {\text { (C) } 30^{\circ}} & {\text { (D) } 60^{\circ}} & {\text { (E) } 120^{\circ}}\end{array}\)

Short Answer

Expert verified
-60 degrees

Step by step solution

01

Identify the value of the inverse tangent

The value we're interested in is \(\tan^{-1}(-\sqrt{3})\). This refers to the angle whose tangent is -鈭�3.
02

Recall tangent values

Remember the tangent values for common angles. The tangent of 60 degrees is 鈭�3, and the tangent of -60 degrees is -鈭�3.
03

Choose the correct answer

The angle whose tangent is -鈭�3 is -60 degrees. Therefore, \(\tan^{-1}(-\sqrt{3})\) is -60 degrees.

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They play a crucial role in simplifying and solving trigonometric expressions and equations. Some of the fundamental identities include:
  • Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
  • Reciprocal Identities: \( \sin(\theta) = \frac{1}{\csc(\theta)} \), \( \cos(\theta) = \frac{1}{\sec(\theta)} \), and \( \tan(\theta) = \frac{1}{\cot(\theta)} \)
  • Quotient Identity: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
These identities are used to find equivalent expressions, reduce complexity, and derive new functions. Their usefulness is evident in exercises involving inverse trigonometric functions. Understanding these identities allows you to manipulate angles and functions efficiently to find specific values or solve equations.
Tangent Function
The tangent function, denoted as \( \tan(\theta) \), is one of the six fundamental trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right triangle. Algebraically, it is expressed through the formula:
  • \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
The tangent function has several unique characteristics:
  • It is periodic, with a period of \(180^{\circ}\) or \(\pi\) radians, which means it repeats every \(180^{\circ}\).
  • It is undefined for angles \(90^{\circ} + k \times 180^{\circ} \) (where \(k\) is an integer) because \( \cos(\theta) \) is zero.
  • The function is also odd, meaning \( \tan(-\theta) = -\tan(\theta) \), which explains why the tangent of \(-60^{\circ}\) is \(-\sqrt{3}\).
The inverse tangent function, \( \tan^{-1}(x) \), finds an angle whose tangent is \(x\). In this case, \( \tan^{-1}(-\sqrt{3}) \) uses the property of the tangent function being odd to match \(-60^{\circ}\) as the solution.
Degrees and Radians
Degrees and radians are the two units used to measure angles. Understanding the relationship between them is essential for solving trigonometric problems and interpreting results correctly.
  • **Degrees**: Commonly used for practical applications because of the easily recognizable range from \(0^{\circ}\) to \(360^{\circ}\).
  • **Radians**: Often used in calculus and higher mathematics due to their natural relation to the circle geometry, defined where \(2\pi\) radians equal \(360^{\circ}\).
The conversion formula between degrees and radians is:
  • \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)
  • \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)
In trigonometry, using the right unit ensures that functions return accurate and meaningful results. For example, while solving \( \tan^{-1}(-\sqrt{3}) \), aligning everything in degrees allowed the direct comparison with options in a multiple-choice question. Being comfortable switching between the two units is a critical skill in mathematical problem-solving.

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Most popular questions from this chapter

extending the idea The Witch of Agnesi The bell-shaped witch of Agnesi can be constructed as follows. Start with the circle of radius \(1,\) centered at the point \((0,1)\) as shown in the figure Choose a point \(A\) on the line \(y=2,\) and connect it to the origin with a line segment. Call the point where the segment crosses the circle \(B .\) Let \(P\) be the point where the vertical line through \(A\) crosses the horizontal line through \(B\) . The witch is the curve traced by \(P\) as \(A\) moves along the line \(y=2\) .Find a parametrization for the witch by expressing the coordinates of \(P\) in terms of \(t\) , the radian measure of the angle that segment OA makes with the positive \(x\) -axis. The following equalities (which you may assume) will help: (i) \(x=A Q \quad\) (ii) \(y=2-A B \sin t \quad\) (iii) \(A B \cdot A O=(A Q)^{2}\)

Transformations Let \(x=(2 \cos t)+h\) and \(y=(2 \sin t)+k\) (a) Writing to Learn Let \(k=0\) and \(h=-2,-1,1,\) and \(2,\) in turn. Graph using the parameter interval \([0,2 \pi]\) . Describe the role of \(h\) (b) Writing to Learn Let \(h=0\) and \(k=-2,-1,1,\) and \(2,\) in turn. Graph using the parameter interval \([0,2 \pi] .\) Describe the role of \(k\) (c) Find a parametrization for the circle with radius 5 and center at \((2,-3)\)(d) Find a parametrization for the ellipse centered at \((-3,4)\) with semi major axis of length 5 parallel to the \(x\) -axis and semi-minor axis of length 2 parallel to the \(y\) -axis (d) Find a parametrization for the ellipse centered at \((-3,4)\) with semi major axis of length 5 parallel to the \(x\) -axis and semi minor axis of length 2 parallel to the \(y\) -axis.

In Exercises 45 and \(46,\) a parametrization is given for a curve. (a) Graph the curve. What are the initial and terminal points, if any? Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? $$x=-\sec t, \quad y=\tan t, \quad-\pi / 2

\(y=|\tan x|\)

Multiple Choice Which of the following is the inverse of \(f(x)=3 x-2 ? \mathrm{}\)
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