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Chapter 6: Problem 62

Find the function value. Round to four decimal places. $$\csc 45.2^{\circ}$$

Short Answer

Expert verified
\(\csc 45.2^{\circ} \approx 1.4092\)

Step by step solution

01

- Understand the Problem

To find the value of \(\csc 45.2^{\circ}\), we need to understand that the cosecant function is the reciprocal of the sine function. Therefore, \(\csc \theta\) is equal to \(\frac{1}{\sin \theta}\).
02

- Find \(\sin 45.2^{\circ}\)

Use a calculator to find the value of \(\sin 45.2^{\circ}\). Make sure your calculator is set to degree mode. The value is \(\sin 45.2^{\circ} \approx 0.7093\).
03

- Calculate the Reciprocal

Since \(\csc 45.2^{\circ}\) is the reciprocal of \(\sin 45.2^{\circ}\), calculate \(\csc 45.2^{\circ} = \frac{1}{0.7093} \approx 1.4092\).

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Key Concepts

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

Cosecant Function
The cosecant function, denoted as \(\text{csc} \theta\), is one of the six fundamental trigonometric functions. The cosecant function is the reciprocal of the sine function. This means that for any angle \(\theta\), the cosecant is calculated as \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\).
The relationship between sine and cosecant is crucial to understanding how to compute values using the trigonometric functions. For instance, if you know the value of sine for a given angle, you can easily find the value of cosecant by taking the reciprocal. This property simplifies many trigonometric computations.
Reciprocal Trigonometric Functions
Reciprocal trigonometric functions are simply the inverse of the primary trigonometric functions. There are three pairs of reciprocal relationships among the six trigonometric functions:
  • \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\)
  • \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\)
  • \(\text{cot} \theta = \frac{1}{\text{tan} \theta}\)
Understanding these reciprocal relationships is important for solving various trigonometric problems. For example, knowing that \(\text{csc} \theta\) is simply the reciprocal of \(\text{sin} \theta\) allows you to compute one if you have the other.
This concept of reciprocal functions applies when working with any angle, whether in degrees or radians. It also helps in understanding and transforming trigonometric equations.
Using a Calculator for Trigonometry
When solving trigonometric problems, calculators can be incredibly helpful. Here鈥檚 how you can use a calculator effectively for trigonometry:
  • Ensure your calculator is set to the correct mode (degree or radian), depending on the angle measurement you are working with. For the exercise given, it should be in degree mode.
  • To find \(\text{sin} 45.2^{\circ}\), simply enter the angle into your calculator and press the sine function button.
  • Once you have the sine value, you can calculate the cosecant by taking its reciprocal. Most calculators do not have a direct cosecant button, so you will need to use the reciprocal function \( \frac{1}{x}\).
For example, to solve the given exercise, first calculate \(\text{sin} 45.2^{\circ} \) which gives approximately 0.7093. Then, use the reciprocal function to find \(\text{csc} 45.2^{\circ} = \frac{1}{0.7093} \) which is approximately 1.4092.
Calculators simplify the process and provide accurate results, which is essential for more complex trigonometric problems.

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Most popular questions from this chapter

Determine the domain of the function. $$f(x)=\frac{\sin x}{\cos x}$$

Given the function value and the quadrant restriction, find \(\theta\). FUNCTION VALUE = \(\tan \theta=-3.0545\) INTERVAL = \(\left(270^{\circ}, 360^{\circ}\right)\) \(\boldsymbol{\theta}\) = ____

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