/*! 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 1 Find the angles \(\sin ^{-1} y\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 1

Find the angles \(\sin ^{-1} y\) and \(\cos ^{-1} y\) and \(\tan ^{-1} y\) in radians. \(y=0\)

Short Answer

Expert verified
\( \sin^{-1}(0) = 0 \), \( \cos^{-1}(0) = \frac{\pi}{2} \), \( \tan^{-1}(0) = 0 \).

Step by step solution

01

Understanding the Problem

We need to find the angles corresponding to inverse trigonometric functions: \( \sin^{-1}(y) \), \( \cos^{-1}(y) \), and \( \tan^{-1}(y) \) where \( y = 0 \). Each of these functions will return an angle measure in radians.
02

Finding \( \sin^{-1}(0) \)

The function \( \sin^{-1}(x) \) returns the angle \( \theta \) whose sine is \( x \), with \( \theta \) in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). Since \( \sin(0) = 0 \), \( \sin^{-1}(0) = 0 \). So, the angle is \( 0 \) radians.
03

Finding \( \cos^{-1}(0) \)

The function \( \cos^{-1}(x) \) returns the angle \( \theta \) whose cosine is \( x \), with \( \theta \) in the range \( [0, \pi] \). Since \( \cos(\frac{\pi}{2}) = 0 \), \( \cos^{-1}(0) = \frac{\pi}{2} \). So, the angle is \( \frac{\pi}{2} \) radians.
04

Finding \( \tan^{-1}(0) \)

The function \( \tan^{-1}(x) \) returns the angle \( \theta \) whose tangent is \( x \), with \( \theta \) in the range \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). Since \( \tan(0) = 0 \), \( \tan^{-1}(0) = 0 \). So, the angle is \( 0 \) radians.

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.

arcsin function
The arcsine function, denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), is the inverse of the sine function. It allows us to find an angle whose sine value we know. Let's make sense of this with an example. Suppose you have a sine value of 0, and you need to determine what angle corresponds to this value in radians. This is precisely what \( \arcsin(0) \) tells us.

  • The answer to \( \arcsin(0) \) is the angle \( \theta \) such that \( \sin(\theta) = 0 \).
  • This angle \( \theta \) must lie within the range \([ -\frac{\pi}{2}, \frac{\pi}{2} ]\), which is the principal range for the arcsin function.
  • Since \( \sin(0) = 0 \), clearly \( \arcsin(0) = 0 \) radians.
Understanding these boundaries is crucial because they tell us the possible values arcsin can return. This ensures that for any input, arcsin gives one unique angle.
arccos function
The arccosine function, expressed as \( \cos^{-1}(x) \) or \( \arccos(x) \), is used to determine the angle whose cosine value is known. By definition, it helps us reverse the cosine function to find the corresponding angle in its principal range, \([0, \pi]\).

Suppose you're given that \( \cos(\theta) = 0 \). The question is, "What is the angle \( \theta \)?" This is where \( \arccos(0) \) comes into play.
  • The principal range for arccos is \([0, \pi]\), meaning it will only give us angles between 0 and \( \pi \) radians.
  • Since \( \cos(\frac{\pi}{2}) = 0 \), this means \( \arccos(0) = \frac{\pi}{2} \).
  • This unique angle ensures we have a consistent output, making the arccos function predictable and reliable.
Grasping the concept of the principal range helps us understand which angles we're typically dealing with when using arccos.
arctan function
The arctangent function, commonly written as \( \tan^{-1}(x) \) or \( \arctan(x) \), helps us find the angle whose tangent is a specific number. This process is vital in situations where we know the tangent of an angle and wish to find the angle itself.

Consider the scenario where \( \tan(\theta) = 0 \). Finding \( \theta \) involves determining \( \arctan(0) \), which is simply asking "What angle has a tangent of 0?"
  • The principal range for arctan is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), indicating that it returns angles within this interval.
  • Since \( \tan(0) = 0 \), \( \arctan(0) = 0 \) radians, which falls within the principal range.
  • Because it covers all real numbers, it's often used in practical applications like finding angles in right triangles.
The principal range ensures each inverse tangent output is distinct and ideal for understanding how functions and angles connect in trigonometry.

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

In \(39-42\) find \(d x / d y\) at the given point. $$ y=\sin x \text { at } x=\pi / 6 $$

Compute the derivatives in exercise.(using the letters as given). $$ u=\sin ^{-1} y / \cos ^{-1} \sqrt{1-y^{2}} $$

36 Suppose \(g(x)\) is continuous at \(x=4,\) say \(g(4)=7 .\) Suppose \(f(y)\) is continuous at \(y=7,\) say \(f(7)=9 .\) Then \(f(g(x))\) is continuous at \(x=4\) and \(f(g(4))=9\). Proof \(\varepsilon\) is given. Because ________ is continuous, there is a \(\delta\) such that \(|f(g(x))-9|<\varepsilon\) whenever \(|g(x)-7|<\delta\). Then because _________ is continuous, there is a \(\theta\) such that \(|g(x)-7|<\delta\) whenever \(|x-4|<\theta,\) Conclusion: If \(|x-4|<\theta\) then ____________\(.\) This shows that \(f(g(x))\) approaches \(f(g(4))\).

We know that \(\sin \pi=0\). Why isn't \(\pi=\sin ^{-1} 0 ?\)

In 31-38 draw the graph of \(y=g(x)\). Separately draw its mirror image \(x=g^{-1}(y)\) $$ y=1 / \sqrt{1-x^{2}}, 0 \leqslant x<1 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.