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Chapter 1: Problem 4

Write each of the following in terms of \(\sin \theta\) only. $$ \csc \theta $$

Short Answer

Expert verified
\(\csc \theta = \frac{1}{\sin \theta}\).

Step by step solution

01

Understand Cosecant in terms of Sine

The cosecant function, denoted as \(\csc \theta\), is the reciprocal of the sine function. This means that \(\csc \theta = \frac{1}{\sin \theta}\). Our task is to express \(\csc \theta\) using only \(\sin \theta\).
02

Express Cosecant as a Reciprocal of Sine

Using the reciprocal identity, we can express \(\csc \theta\) in terms of \(\sin \theta\): \[\csc \theta = \frac{1}{\sin \theta}\]Thus, we have written the expression in terms of \(\sin \theta\) only.

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Key Concepts

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

Reciprocal Identity
A core concept in trigonometry is the idea of reciprocal identities, which relate one function to another through reciprocals. In this context, the reciprocal identity pertains to sine and cosecant functions. If you're not familiar with the term, a reciprocal simply means one divided by that function. For instance, if you have a number like 2, its reciprocal is \( \frac{1}{2} \).
  • In trigonometric terms, the reciprocal identity for sine, denoted as \( \sin \theta \), is \( \csc \theta = \frac{1}{\sin \theta} \).
  • This identity is immensely useful because it allows us to switch perspective on a trigonometric function. If we know the value of one, we can easily find its reciprocal.
Understanding reciprocal identities can simplify solving problems involving trigonometric functions, as it provides a straightforward way to transition between different functions using known values.
Cosecant Function
The cosecant function is an essential trigonometric function that often appears in various mathematical problems. Represented as \( \csc \theta \), its primary relationship is with the sine function, as it is its reciprocal. This means that the cosecant function focuses on the inverse of what sine represents.
  • Mathematically, it is expressed as \( \csc \theta = \frac{1}{\sin \theta} \).
  • This function is undefined when \( \sin \theta = 0 \) because division by zero is not possible in mathematics.
  • The graph of the cosecant function will generally show vertical asymptotes wherever the sine value is zero, signaling these points of indefiniteness.
Analyzing the cosecant function by its reciprocal identity with sine helps in understanding and visualizing its properties more effectively.
Sine Function
The sine function is one of the foundational trigonometric functions originating from the study of right triangles. In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. It's commonly written as \( \sin \theta \).
  • In a unit circle context, \( \sin \theta \) is the y-coordinate of a point on the circle.
  • Its range is always between -1 and 1, inclusive.
  • Sine is a periodic function with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians.
Understanding the sine function is crucial, as it is a gateway to learning about its reciprocal, cosecant, and further trigonometric identities. Its interplay between geometric and algebraic properties offers deep insight into various aspects of mathematics and applied sciences.

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Most popular questions from this chapter

Give the reciprocal of each number. 7

Find the remaining trigonometric ratios of \(\theta\) based on the given information. \(\cos \theta=\frac{2 \sqrt{13}}{13}\) and \(\theta \in \mathrm{QIV}\)

For Problems 27 through 30 , let \(\sin \theta=-12 / 13\), and \(\cos \theta=-5 / 13\), and find the indicated value. \(\tan \theta\)

Use the reciprocal identities for the following problems. If \(\csc \theta=-\frac{13}{5}\), find \(\sin \theta\)

For Problems 23 through 26 , recall that \(\sin ^{2} \theta\) means \((\sin \theta)^{2}\). If \(\cos \theta=\frac{\sqrt{3}}{2}\), find \(\cos ^{2} \theta\)
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