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Chapter 4: Problem 43

Evaluate the trigonometric function of the quadrant angle. $$ \csc \pi $$

Short Answer

Expert verified
\(\csc \pi\) is undefined.

Step by step solution

01

Understand the Function

The Cosecant (\(\csc\)) function is defined as the reciprocal of sine function. It is denoted as \(\csc \theta = 1/\sin \theta\). However, it's important to notate that the cosecant function is undefined where the sine function is 0.
02

Determine the Value of Sine Function at \(\pi\)

The sine value of \(\pi\) radian (180 degrees) is 0. This is because when the angle is at \(\pi\) radians, it lies on the negative x-axis, and the y-coordinate (which represents \(\sin\) value) of the point on the unit circle is 0.
03

Calculate the Cosecant Value

Cosecant is the reciprocal of sine. Therefore, to calculate the cosecant value we take the reciprocal of sine. However, the reciprocal of 0 is undefined. Therefore, \(\csc \pi\) is undefined.

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Key Concepts

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

Cosecant
Cosecant is one of the six fundamental trigonometric functions, often abbreviated as \(\csc\). It is the reciprocal of the sine function. Mathematically, it is expressed as \(\csc \theta = \frac{1}{\sin \theta}\). However, there's an important condition: cosecant is undefined whenever the sine function equals zero. Why? Because you can't divide by zero in mathematics.
  • If \(\sin \theta = 0\), then \(\csc \theta\) becomes undefined.
  • Typical angles where sine equals zero include 0, \(\pi\), 2\(\pi\), and so on.
Understanding when a trigonometric function is undefined is crucial for solving trigonometric problems accurately and avoiding errors in calculations.
Sine Function
The sine function is a foundational trigonometric function with the notation \(\sin \theta\). It gives the y-coordinate of a point on the unit circle for a given angle \(\theta\). The range of the sine function is between -1 and 1. Let's discuss some key points:
  • The value of \(\sin \theta\) directly relates to the height of the point when projected onto the y-axis.
  • At an angle of \(\pi\) radians, or 180 degrees, \(\sin \pi = 0\).
  • This occurs because the point at this angle lies exactly on the negative x-axis with no height, rendering the sine value zero.
Understanding the basic values of sine at key angles like 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\) is foundational for mastering trigonometry.
Unit Circle
The unit circle is an essential concept in trigonometry and is a circle with a radius of 1 centered at the origin of a coordinate plane. It serves as the basis for defining the trigonometric functions. Here are some important aspects:
  • The equation of the unit circle is given by \(x^2 + y^2 = 1\).
  • Angles on the unit circle are measured from the positive x-axis, moving counterclockwise.
  • At any point on the unit circle, the coordinates correspond to \(\cos \theta\) for the x-value and \(\sin \theta\) for the y-value.
  • Key points are \( (1, 0) \), \( (0, 1) \), \((-1, 0) \), and \( (0, -1)\), representing angles \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\) respectively.
The unit circle not only visualizes angles and their trigonometric values but also provides an intuitive way to understand the periodic nature of these functions.

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Most popular questions from this chapter

In Exercises 49-54, use the properties of inverse trigonometric functions to evaluate the expression. \(sin(arcsin\ 0.3)\)

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