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Chapter 2: Problem 66

Given that \(\csc x\) is undefined, what is the value of \(\sin x ?\) Explain.

Short Answer

Expert verified
The value of \(\sin x\) is 0.

Step by step solution

01

Understand the Cosecant Function

The cosecant function, \(\csc x\), is the reciprocal of the sine function. This means that \(\csc x = \frac{1}{\sin x}\). To determine when \(\csc x\) is undefined, consider when the denominator \((\sin x)\) is zero.
02

Identify When Cosecant is Undefined

\(\csc x\) is undefined when \( rac{1}{\sin x}\) does not exist. This occurs precisely when \(\sin x = 0\), as division by zero is undefined.
03

Analyze the Meaning of \\( ext{sin}\\) Being Zero

Since \(\sin x = 0\) results in \(\csc x\) being undefined, we conclude that the value of \(\sin x\) when \(\csc x\) is undefined is 0.

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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 \(\csc x\), is one of the basic trigonometric functions. It is rarely introduced alone; rather, it's often presented in relation to the sine function because it is, in fact, the sine's reciprocal.
In simpler terms, for any given angle \(x\), \(\csc x\) is calculated as \(\csc x = \frac{1}{\sin x}\).
It's essential to grasp this relationship because it tells us a lot about the behavior of \(\csc x\) in different scenarios. Knowing this will also help us understand situations where \(\csc x\) becomes undefined鈥攕omething students encounter quite often.
So whenever you come across \(\csc x\), just remember it ties directly back to the sine function.
Sine Function
The sine function, represented as \(\sin x\), is one of the most fundamental trigonometric functions and is essential for understanding the behavior of angles and triangles. It tells us a coordinate value based on an angle in a unit circle.
When calculating \(\sin x\), it essentially gives the vertical position on a circle when the angle \(x\) is measured from the positive x-axis. The outputs of \(\sin x\) range between -1 and 1.
This function is a vital building block of trigonometry. Knowing when its value is zero helps us conclude when its reciprocal, the \(\csc x\), becomes undefined.
Undefined Expression
An undefined expression in mathematics occurs when you attempt to calculate something that doesn't have a meaningful answer. In trigonometry, this often happens when trying to divide by zero.
For instance, \(\csc x = \frac{1}{\sin x}\) becomes undefined precisely when \(\sin x = 0\) because division by zero is not valid. The concept of undefined expressions is crucial in mathematics as it sets boundaries on what values functions can or cannot take.
Whenever you encounter a scenario where something appears undefined, check the denominator of fractions, as such cases often hint at division by zero.

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Most popular questions from this chapter

Evaluate the trigonometric expressions with a calculator. Round your answer to four decimal places. $$ \csc \left(-111^{\circ}\right) $$

Evaluate the following expressions exactly by using a reference angle. $$ \tan 210^{\circ} $$

Find the smallest possible positive measure of \(\theta\) (rounded to the nearest degree) if the indicated information is true. \(\cos \theta=-0.7986\) and the terminal side of \(\theta\) lies in quadrant II.

Evaluate the following expressions exactly by using a reference angle. $$ \cot \left(-150^{\circ}\right) $$

Evaluate the expression sec \(120^{\circ}\) exactly. Solution: \(120^{\circ}\) lies in quadrant II. The reference angle is \(30^{\circ}\). Find the cosine of the reference angle. $$ \cos 30^{\circ}=\frac{\sqrt{3}}{2} $$ Cosine is negative in quadrant II. \(\cos 120^{\circ}=-\frac{\sqrt{3}}{2}\) Secant is the reciprocal of cosine. \(\sec 120^{\circ}=-\frac{2}{\sqrt{3}}=-\frac{2 \sqrt{3}}{3}\) This is incorrect. What mistake was made?
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