/*! 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 91 Determine the domain and the ran... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 91

Determine the domain and the range of each function. $$ f(x)=\sin ^{-1} x+\cos ^{-1} x $$

Short Answer

Expert verified
The domain of the function \(f(x) = \sin^{-1}(x) + \cos^{-1}(x)\) is [-1, 1], and the range is \(\{\frac{\pi}{2}\}\).

Step by step solution

01

Identify the Domain of Individual Functions

The domain of \(\sin^{-1}(x)\) and \(\cos^{-1}(x)\) is [-1, 1] because any x-value outside of this range would make the functions nonreal. Hence, this will be the domain of the whole function.
02

Identify the Range of Individual Functions

The range of \(\sin^{-1}(x)\) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) and the range of \(\cos^{-1}(x)\) is from 0 to \(\pi\). Therefore, the smallest possible value for the function \(f(x) = \sin^{-1}(x) + \cos^{-1}(x)\) occurs when \(f(x) = \sin^{-1}(-1) + \cos^{-1}(-1) = -\frac{\pi}{2} + \pi = \frac{\pi}{2}\). And the highest possible value occurs when \(f(x) = \sin^{-1}(1) + \cos^{-1}(1) = \frac{\pi}{2} + 0 = \frac{\pi}{2}\). Thus, the range of the given function is only one value, which is \(\frac{\pi}{2}\).
03

Compile the Results

The domain and range of a function can be given in interval notation. Now, compile these results to derive the domain and range of the entire function. The domain of \(f(x) = \sin^{-1}(x) + \cos^{-1}(x)\) is [-1, 1], and the range is \(\{\frac{\pi}{2}\}\) as every input gives the same output.

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.

Inverse Trigonometric Functions
Inverse trigonometric functions are essential in mathematics to reverse the effect of standard trigonometric functions like sine, cosine, and tangent. They allow us to find angles when given the ratio of the sides of a right triangle.
  • The most common inverse trigonometric functions are arcsine, arccosine, and arctangent.
  • These functions are written as \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \tan^{-1}(x) \).
  • They map specific intervals of real numbers back to angles.
  • Each inverse function has its own specific domain and range.
Understanding where these functions are applicable and their limitations is crucial for solving equations and interpreting real-world situations effectively. By identifying the domain and range, it becomes easier to work with these functions in mathematical problems.
Arcsine
The arcsine function, written as \( \sin^{-1}(x) \), is the inverse of the sine function. It provides the angle whose sine value equals a given number.
  • The domain of arcsine is restricted to the interval \([-1, 1]\), meaning it only accepts input values within this range.
  • Its range is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), so the output is always an angle within these bounds.
  • When solving for \( \theta = \sin^{-1}(x) \), remember that \( \theta \) is the angle whose sine is \( x \).
This function is particularly useful in geometry and physics when determining angles based on height and distance ratios. It's important to know its range and domain to avoid errors in calculations.
Arccosine
The arccosine function, denoted as \( \cos^{-1}(x) \), is the inverse of the cosine function. It provides the angle whose cosine corresponds to a particular number.
  • Just like arcsine, arccosine is defined for values within the domain \([-1, 1]\).
  • Its range spans from \(0\) to \(\pi\), which ensures outputs are angles measured in radians.
  • If \( \theta = \cos^{-1}(x) \), \( \theta \) represents the angle where the cosine of \( \theta \) is equal to \( x \).
Understanding arccosine is important when analyzing situations involving horizontal distances or projections. By knowing its domain and range, students can better grasp its applications in various mathematical and engineering contexts.

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

Explain why, without restrictions, no trigonometric function has an inverse function.

Your neighborhood movie theater has a 25 -foot-high screen located 8 feet above your eye level. If you sit too close to the screen, your viewing angle is too small, resulting in a distorted picture. By contrast, if you sit too far back, the image is quite small, diminishing the movie's visual impact. If you sit \(x\) feet back from the screen, your viewing angle, \(\theta,\) is given by $$\theta=\tan ^{-1} \frac{33}{x}-\tan ^{-1} \frac{8}{x}$$ (GRAPH CANNOT COPY) Find the viewing angle, in radians, at distances of 5 feet, 10 feet, 15 feet, 20 feet, and 25 feet.

Graph each pair of functions in the same viewing rectangle. Use your knowledge of the domain and range for the inverse trigonometric function to select an appropriate viewing rectangle. How is the graph of the second equation in cach exercise related to the graph of the first equation? $$ y=\sin ^{-1} x \text { and } y=\sin ^{-1}(x+2)+1 $$

a. Graph the restricted cotangent function, \(y=\cot x,\) by restricting \(x\) to the interval \((0, \pi)\). b. Use the horizontal line test to explain why the restricted cotangent function has an inverse function. c. Use the graph of the restricted cotangent function to graph \(y=\cot ^{-1} x\).

If \(\theta=\frac{3}{2},\) is this angle larger or smaller than a right angle?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

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.