/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 Writing Explain why each express... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 57

Writing Explain why each expression is undefined. $$ \sec 90^{\circ} $$

Short Answer

Expert verified
Sec(90 degrees) is undefined because it equals 1 divided by cos(90 degrees), which equals 1 divided by 0, and division by zero in mathematics is undefined.

Step by step solution

01

Understanding secant

The secant of an angle is the reciprocal of the cosine of that angle. Therefore, when we see sec(90 degrees), it's basically 1 / cos(90 degrees).
02

Calculating the cosine of 90 degrees

Next, we calculate the cosine of 90 degrees. From trigonometric ratios, we know that cos(90 degrees) = 0.
03

Explaining why sec(90 degrees) is undefined

Now we apply the value of cos(90 degrees) back into the original equation sec(90 degrees) = 1 / cos(90 degrees). This gives us 1 / 0. In mathematics, division by zero is undefined hence making sec(90 degrees) undefined.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Trigonometric Functions
Trigonometric functions are fundamental in mathematics, helping us describe relationships in right-angled triangles and periodic patterns in circles. Common trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent. These functions are key in connecting angles to the ratios of the sides in a triangle, and they also extend to describe wave behaviors in the context of circles.
  • Sine \(\sin\) represents the ratio of the opposite side to the hypotenuse.
  • Cosine \(\cos\) represents the adjacent side to the hypotenuse ratio.
  • Tangent \(\tan\) is the ratio of the opposite side to the adjacent side.
Each function has unique properties and is related through specific identities, as we'll explore in this article.
Reciprocal Identities
Reciprocal identities are handy relationships between trigonometric functions that express one function in terms of another. These identities help simplify complex calculations by allowing for alternate expressions.
The reciprocal identities are:
  • \(\csc \theta = \frac{1}{\sin \theta}\)
  • \(\sec \theta = \frac{1}{\cos \theta}\)
  • \(\cot \theta = \frac{1}{\tan \theta}\)
For secant, this means that secant of an angle is just one over the cosine of that angle. Using these relationships can easily transform the way we solve equations and provide insight into undefined expressions.
Division by Zero
Division by zero is a mathematical expression that essentially breaks the rules. When we try to divide a number by zero, the operation doesn't yield a valid result, because there's no number that you can multiply by zero to get something other than zero.
Thus, any time you find yourself with a calculation like \(\frac{1}{0}\), it becomes undefined. Undefined expressions don't have a meaningful interpretation in mathematics.
This concept crucially informs why sec(90掳) is undefined since it leads to division by zero when evaluated using reciprocal identities.
Secant Function
The secant function, denoted as \(\sec\), is one of the trigonometric functions and is specifically the reciprocal of cosine. It holds unique characteristics and undefined values at certain points due to its dependence on cosine.
For instance, \(\sec 90^\circ = \frac{1}{\cos 90^\circ}\). Since \(\cos 90^\circ = 0\), this results in division by zero. Thus, secant is undefined at 90掳 and similar angles where cosine equals zero.
Understanding these specifics helps avoid errors in calculations and gives better insights into the behaviors and limitations of the secant function.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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

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

Find the amplitude and period of each function. Describe any phase shift and vertical shift in the graph. $$ y=-\cos (x+4)-7 $$

Evaluate each expression. Write your answer in exact form. Suppose \(\tan \theta=\frac{20}{15} .\) Find \(\cot \theta\)

Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places. $$ \csc 70^{\circ} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.