/*! 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 28 Find the exact value of each of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 28

Find the exact value of each of the following expressions without using a calculator. $$\csc \left(240^{\circ}\right)$$

Short Answer

Expert verified
The exact value of \( \text{csc}(240^{\text{掳}}) = -\frac{2\text{鈭�3}}{3} \).

Step by step solution

01

Identify the related angle in the unit circle

The angle given is \(240^{\text{掳}}\). Since the unit circle completes one full cycle at 360掳, calculating \(240^{\text{掳}}\) is straightforward. \(240^{\text{掳}}\) is in the third quadrant.
02

Determine the reference angle

In the third quadrant, the reference angle \( \theta_{ref} \) for \(240^{\text{掳}}\) is obtained by subtracting \(180^{\text{掳}}\). Therefore, \(240^{\text{掳}} - 180^{\text{掳}} = 60^{\text{掳}}\).
03

Find the sine of the reference angle

Know that \( \text{sin}(60^{\text{掳}}) = \frac{\text{鈭�3}}{2} \). Also, since \(240^{\text{掳}}\) is in the third quadrant and sine is negative there, \(\text{sin}(240^{\text{掳}}) = -\frac{\text{鈭�3}}{2} \).
04

Calculate the cosecant

The cosecant function is the reciprocal of the sine function. Thus, \(\text{csc}(240^{\text{掳}}) = \frac{1}{\text{sin}(240^{\text{掳}})}\). By substituting, \( \text{csc}(240^{\text{掳}}) = \frac{1}{-\frac{\text{鈭�3}}{2}} = -\frac{2}{\text{鈭�3}} \).
05

Rationalize the denominator

To rationalize \( -\frac{2}{\text{鈭�3}} \), multiply both the numerator and the denominator by \( \text{鈭�3} \) to get \(-\frac{2 \times \text{鈭�3}}{3} = -\frac{2\text{鈭�3}}{3} \).

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.

Unit Circle
The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane. This circle helps to define trigonometric functions for all angles.
Within the unit circle:
  • Angles are measured starting from the positive x-axis and move counterclockwise.
  • Each angle has coordinates on the circle corresponding to \( ( \text{cos}(\theta), \text{sin}(\theta) ) \), where \( \theta \) is the angle in radians or degrees.
  • The unit circle is divided into four quadrants, with specific angle ranges for each.
For example, the angle \(240^{\circ}\) is located in the third quadrant of the unit circle. Understanding the position of angles in the unit circle is crucial for determining the values of trigonometric functions.
Reference Angle
Reference angles are useful for simplifying trigonometry problems. They are the smallest angle that a given angle makes with the x-axis.
Here's how to find the reference angle for any given angle:
  • For angles in the first quadrant, the reference angle is the same as the original angle.
  • In the second quadrant, subtract the angle from \(180^{\circ}\).
  • In the third quadrant, subtract \(180^{\circ}\) from the angle.
  • In the fourth quadrant, subtract the angle from \(360^{\circ}\).
For instance, for \(240^{\circ}\) in the third quadrant, the reference angle is calculated as \(240^{\circ} - 180^{\circ} = 60^{\circ}\). This reference angle helps us find trigonometric values conveniently.
Cosecant Function
The cosecant function, denoted as \( \text{csc}(\theta) \), is the reciprocal of the sine function. This means:
\[ \text{csc}(\theta) = \frac{1}{\text{sin}(\theta)} \]
  • We use the cosecant function to find out the ratio of the hypotenuse to the opposite side in a right triangle.
  • It is undefined for angles where the sine is zero (e.g., \(0^{\circ}, 180^{\circ}, 360^{\circ}\)).
For a given angle like \(240^{\circ}\), first, find \( \text{sin}(240^{\circ}) = -\frac{\text{鈭�3}}{2} \). Then, the cosecant is found by taking the reciprocal: \( \text{csc}(240^{\circ}) = \frac{1}{-\frac{\text{鈭�3}}{2}} = -\frac{2}{\text{鈭�3}} \).
Rationalizing Denominators
Rationalizing the denominator involves modifying a fraction so that the denominator becomes a rational number (i.e., no square roots).
Here鈥檚 the step-by-step process:
  • Take the fraction \( -\frac{2}{\text{鈭�3}} \) which needs rationalizing.
  • Multiply the numerator and the denominator by \( \text{鈭�3} \), which gives:
    \[ -\frac{2}{\text{鈭�3}} \times \frac{\text{鈭�3}}{\text{鈭�3}} = -\frac{2 \times \text{鈭�3}}{\text{鈭�3} \times \text{鈭�3}} = -\frac{2 \text{鈭�3}}{3} \]
Now, the denominator is a rational number. Rationalizing denominators is important for standardized math conventions and simplifying calculations.

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

Determine the period and sketch at least one cycle of the graph of each function. $$y=\cot (x / 2)$$

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\csc (x-\pi / 2)$$

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\sec (\pi x / 2)$$

The velocity \(v\) of blood at a valve in the heart of a certain rodent is modeled by the equation \(v=-4 \cos (6 \pi t)+4,\) where \(v\) is in centimeters per second and \(t\) is time in seconds. a. What are the maximum and minimum velocities of the blood at this valve? b. What is the rodent's heart rate in beats per minute?

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\frac{1}{2} \sec x$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.