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Chapter 8: Problem 54

Find the exact value of each expression. $$ \csc \left[\tan ^{-1}(-2)\right] $$

Short Answer

Expert verified
- \(\frac{\sqrt{5}}{2}\)

Step by step solution

01

Understand Inverse Trigonometric Function

First, recognize that \(\tan^{-1}(-2)\) represents an angle whose tangent is \(-2\). Denote this angle by \(\theta\); thus, \( \theta = \tan^{-1}(-2) \) and \(\tan(\theta) = -2\).
02

Use a Right Triangle

To find \(\theta\), imagine a right triangle where \(\tan(\theta)\) is given by the ratio of the opposite side to the adjacent side. Since \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = -2\), let the opposite side be \(-2\) and the adjacent side be \1\.
03

Calculate the Hypotenuse

Use the Pythagorean theorem to determine the hypotenuse: \(\text{hypotenuse} = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}\).
04

Find the Sine of \( \theta \)

Recall the definition of \( \text{cosecant}\): \( \text{cosecant} \theta = \frac{1}{\text{sine} \theta} \). First, find the sine of \( \theta \): \( \text{sine} \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-2}{\sqrt{5}}\).
05

Simplify the Expression

Therefore, \( \text{csc}[\tan^{-1}(-2)] = \frac{1}{\text{sine} \theta} = \frac{1}{\frac{-2}{\sqrt{5}}} = -\frac{\sqrt{5}}{2}\).

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Key Concepts

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

inverse trigonometric functions
Inverse trigonometric functions help us find angles when we know their trigonometric ratios. For instance, \(\tan^{-1}(-2)\) signifies the angle whose tangent is \(-2\). If you see \(\theta = \tan^{-1}(-2)\), it means \(\tan(\theta) = -2\). Each inverse function operates within a specific range to give a unique angle. This helps in determining angles corresponding to given trigonometric values. In our example, this is the crucial first step to moving forward with the problem.
right triangle relationships
Working with right triangles simplifies trigonometric calculations. When given a ratio like \(\tan(\theta) = -2\), it tells us the relationship between the sides of the triangle. In this case, for a right triangle representing this tangent ratio, the opposite side could be \-2\ and the adjacent side \1\. This forms the basis for calculating hypotenuse using the Pythagorean theorem.
Right triangles help relate various trigonometric functions through simple geometric relationships.
Pythagorean theorem
The Pythagorean theorem is vital for finding unknown sides in right triangles: \[a^2 + b^2 = c^2\]. Here, \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse. For our triangle with sides \(-2\) and \1\, the hypotenuse \(c\) is calculated as follows:
\[c = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}\].
This hypotenuse helps us find the sine, and subsequently, other trigonometric values for the angle \(\theta\).
cosecant
Cosecant is the reciprocal of sine, i.e., \(\text{csc } \theta = \frac{1}{\text{sin } \theta}\). To solve for \(\text{csc} [\tan^{-1}(-2)]\), first find \(\text{sin } \theta\) when \(\theta = \tan^{-1}(-2)\):
\[ \text{sin } \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-2}{\text{\sqrt{5}}}\].
Take its reciprocal to get the cosecant:
\[ \text{csc}[\tan^{-1}(-2)] = \frac{1}{\frac{-2}{\sqrt{5}}} = -\frac{\sqrt{5}}{2}\].
Understanding these relationships is key in solving similar trigonometric problems.

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Most popular questions from this chapter

Write each trigonometric expression as an algebraic expression containing u and \(v .\) Give the restrictions required on \(u\) and \(v\). $$ \tan \left(\sin ^{-1} u-\cos ^{-1} v\right) $$

Is the function \(f(x)=\frac{3 x}{5-x^{2}}\) even, odd, or neither?

Movie Theater Screens Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, \(\theta\) is the viewing angle. Suppose that you sit \(x\) feet from the screen. The viewing angle \(\theta\) is given by the function $$ \theta(x)=\tan ^{-1}\left(\frac{34}{x}\right)-\tan ^{-1}\left(\frac{6}{x}\right) $$ (a) What is your viewing angle if you sit 10 feet from the screen? 15 feet? 20 feet? (b) If there are 5 feet between the screen and the first row of seats and there are 3 feet between each row and the row behind it, which row results in the largest viewing angle? (c) Using a graphing utility, graph $$ \theta(x)=\tan ^{-1}\left(\frac{34}{x}\right)-\tan ^{-1}\left(\frac{6}{x}\right) $$ What value of \(x\) results in the largest viewing angle?

Solve each equation on the interval \(0 \leq \theta<2 \pi\). $$ \sin \theta-\cos \theta=-\sqrt{2} $$

Write each trigonometric expression as an algebraic expression containing u and \(v .\) Give the restrictions required on \(u\) and \(v\). $$ \sec \left(\tan ^{-1} u+\cos ^{-1} v\right) $$
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