/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 121 In Exercises \(119-130\), assume... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 121

In Exercises \(119-130\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. $$ \operatorname{arcsec}\left(\sec \left(\frac{5 \pi}{6}\right)\right) $$

Short Answer

Expert verified
The exact value is \( \frac{5\pi}{6} \).

Step by step solution

01

Identify the Expression

We are given the expression \( \operatorname{arcsec}(\sec(\frac{5\pi}{6})) \). The task is to find the exact value of this expression.
02

Calculate \(\sec\left(\frac{5\pi}{6}\right)\)

First, we calculate \( \sec(\frac{5\pi}{6}) \). The secant function is the reciprocal of the cosine function. The cosine of \( \frac{5\pi}{6} \) is \( -\frac{\sqrt{3}}{2} \) since it is in the second quadrant where cosine is negative. So, \( \sec\left(\frac{5\pi}{6}\right) = \frac{1}{\cos\left(\frac{5\pi}{6}\right)} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \) after rationalizing the denominator.
03

Apply \(\operatorname{arcsec}\) Function

Next, we apply the arcsecant function. \( \operatorname{arcsec}(x) \) is the angle \( \theta \) such that \( \sec(\theta) = x \), and \( \theta \) is in the range \( \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right] \). We need to find an angle in this range for which the secant is \( -\frac{2\sqrt{3}}{3} \). Since \( \sec(\frac{5\pi}{6}) = -\frac{2\sqrt{3}}{3} \), \( \operatorname{arcsec}(-\frac{2\sqrt{3}}{3}) = \frac{5\pi}{6} \) because \( \frac{5\pi}{6} \) is in the valid range for arcsecant.
04

Verify the Range

Verify that \( \frac{5\pi}{6} \) indeed falls within the specified range \( \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right] \). The angle \( \frac{5\pi}{6} \) is between \( \frac{\pi}{2} \) and \( \pi \), thus satisfying the range requirement.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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 the angles of a right triangle when we know the values of the trigonometric ratios. This reverses what the regular trigonometric functions do. They include functions like arcsine, arccosine, and arcsecant. If you apply an inverse trigonometric function to a number, you get an angle as a result. For example, if you apply the arcsecant (\(\operatorname{arcsec}\)) to a secant value, it gives you the angle whose secant is that number.
  • Inverse functions allow us to move back and forth between angles and trigonometric ratios.
  • They are helpful in determining exact angles in problems involving trigonometry.
Be mindful when using inverse functions as each has its specific range to ensure unique outputs.
Secant Function
The secant function (\(\sec\)) is one of the six primary trigonometric functions and is the reciprocal of the cosine function. \[ \sec(\theta) = \frac{1}{\cos(\theta)} \] This means that if you know the cosine of an angle, you can find the secant by taking the reciprocal. This function is particularly useful when dealing with trigonometric identities and in cases when the cosine value is zero, the secant function is undefined because dividing by zero is not possible.
  • It translates horizontal distances on the unit circle to their reciprocals.
  • Unlike cosine, secant function values can be greater or less than one.
Keep in mind that the secant function can take positive values in the first and fourth quadrants and negative values in the second and third quadrants.
Trigonometric Identities
In trigonometry, identities express relationships between different trig functions and are true for all angle measures. These identities are essential tools in math as they simplify complex trigonometric expressions and assist in solving trigonometric equations. For example, using the identity for secant and cosine, \[ \sec(\theta) = \frac{1}{\cos(\theta)} \] helps to evaluate trigonometric functions or convert expressions from one form to another. Another one involves Pythagorean identities like \[ 1 + \tan^2(\theta) = \sec^2(\theta) \] which can be incredibly handy.
  • They are used to prove new formulas and derive solutions to trigonometric problems.
  • Recognizing and using identities allow for the simplification and solving of problems effortlessly.
Familiarity with these identities can help streamline the process of solving complex mathematical problems.
Angle Ranges
In trigonometry, understanding the range of angles for which functions are defined is crucial, especially with inverse trigonometric functions. Each inverse function has a principal value range to ensure that each function yields unique output. For instance, the arcsecant function has a range of \[ \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right] \] ensuring the value corresponds to a unique angle where the secant is defined.
  • This is important for maintaining the function as a true inverse, which is truly one-to-one.
  • Knowing these ranges helps validate solutions and ensures they meet problem constraints.
By understanding these angle ranges, it's easier to verify solutions and apply the correct function attributes in trigonometric problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(155-164\), find the exact value or state that it is undefined. $$ \cos \left(2 \operatorname{arcsec}\left(\frac{25}{7}\right)\right) $$

In Exercises \(119-130\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. $$ \operatorname{arccsc}\left(\csc \left(\frac{9 \pi}{8}\right)\right) $$

In Exercises \(208-210\), find the two acute angles in the right triangle whose sides have the given lengths. Express your answers using degree measure rounded to two decimal places. $$ 3,4 \text { and } 5 $$

In Exercises \(119-130\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. $$ \operatorname{arccsc}\left(\csc \left(\frac{2 \pi}{3}\right)\right) $$

In Exercises \(155-164\), find the exact value or state that it is undefined. $$ \sin \left(\arcsin \left(\frac{5}{13}\right)+\frac{\pi}{4}\right) $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.