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Chapter 13: Problem 56

Writing Explain why each expression is undefined. $$ \csc 180^{\circ} $$

Short Answer

Expert verified
The expression \(\csc(180^{\circ})\) is undefined because it results in division by zero, which is undefined in mathematics.

Step by step solution

01

Understand the cosecant function

The cosecant function, often written as \(\csc(\theta)\), is defined as the reciprocal of the sine function: \(\csc(\theta) = \frac{1}{\sin(\theta)}\). Now, the sine function ranges between -1 and 1, and certain angles give a sine value of 0 (example: 0掳, 180掳, 360掳, and so on).
02

Analyze the given angle

In this case, the angle given is 180掳. The sine of 180掳 is 0. This comes from the unit circle representation, where at 180掳, the y-coordinate (which is representative of the sine value) is 0.
03

Reason why the expression is undefined

Substituting the value of \(\sin(180^{\circ})\) into the equation of \(\csc(\theta) = \frac{1}{\sin(\theta)}\) gives \(\csc(180^{\circ}) = \frac{1}{0}\). Division by zero is undefined in mathematics, so \(\csc(180^{\circ})\) is undefined.

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Key Concepts

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

Understanding the Sine Function
The sine function is a fundamental part of trigonometry. It's represented as \( \sin(\theta) \), where \( \theta \) is an angle. Sine measures the vertical component or the y-coordinate of a point on the unit circle.
  • It oscillates between -1 and 1 as \( \theta \) varies.
  • On the unit circle, sine reaches its maximum value of 1 at 90掳 and a minimum value of -1 at 270掳.
  • When \( \theta = 180掳 \), the sine value is 0 because the point on the unit circle aligns with the x-axis.
Recognizing these key values and how they relate to different angles is crucial for solving many trigonometric problems.
Exploring Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for every valid input. These identities help us simplify and manipulate trigonometric expressions.
  • Reciprocal identities are highly useful. For instance, the cosecant function is the reciprocal of the sine function: \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
  • Other fundamental identities include \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
These identities are essential tools for solving trigonometric problems and functions, especially when dealing with undefined expressions or complex calculations.
Understanding the Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It's a crucial concept in trigonometry used for defining the trigonometric functions.
  • Each point on the unit circle can be represented as \( (\cos(\theta), \sin(\theta)) \).
  • At 0掳 and 360掳, the coordinate is (1, 0), where \( \sin \) is 0.
  • At 90掳, the coordinate is (0, 1), where \( \sin \) is 1.
  • At 180掳, the coordinate is (-1, 0), where \( \sin \) is 0 once again.
This concept helps visualize how angles correspond to specific sine and cosine values, which makes it easier to comprehend expressions and their evaluations.
Explaining Undefined Expressions
An expression becomes undefined when its mathematical evaluation leads to a scenario that doesn't make sense. This often occurs in division by zero situations.
  • When calculating \( \csc(180掳) \), we find that \( \sin(180掳) = 0 \).
  • Substituting in the reciprocal identity, \( \csc(180掳) = \frac{1}{0} \), results in an undefined expression.
  • Division by zero is undefined because there's no number that can multiply with zero to produce one.
Understanding when and why expressions are undefined is crucial in mathematics, as it helps prevent errors in calculations and interpretations.

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Most popular questions from this chapter

Graph each function in the interval from 0 to 2\(\pi\) $$ y=-\sec \pi \theta $$

a. Graph \(y-\tan x\) and \(y-\cot x\) on the same axes. b. State the domain, the range, and the asymptotes of each function. c. Writing Compare the two graphs. How are they alike? How are they different? d. Geometry The graph of the cotangent function can be reflected about a line to graph the tangent function. Name at least two lines that have this property.

Graph each function in the interval from 0 to 2\(\pi\) $$ y=\csc \theta-\frac{\pi}{2} $$

Graph each function in the interval from 0 to 2\(\pi .\) Describe any phase shift and vertical shift in the graph. $$ f(x)=3 \csc (x+2)-1 $$

Evaluate each expression to the nearest hundredth. Each angle is given in radians. $$ \cot 3 $$
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