/*! 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 64 Prove the following identities. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 64

Prove the following identities. $$\sin ^{-1} y+\sin ^{-1}(-y)=0$$

Short Answer

Expert verified
Question: Prove the identity \(\sin^{-1}y + \sin^{-1}(-y) = 0\). Short Answer: To prove the given identity, we can use the properties of the inverse sine function and the fact that the sine function is an odd function. Since we know that \(\sin(-x) = -\sin(x)\), we can use this property to show that \(-\sin^{-1}(\sin(x)) = \sin^{-1}(-y)\). This results in \(-\sin^{-1}(y) = \sin^{-1}(-y)\), which helps us confirm the identity, as \(\sin^{-1}(y) + \sin^{-1}(-y) = \sin^{-1}(y) - \sin^{-1}(y) = 0\).

Step by step solution

01

Understand inverse sine function

The inverse sine function, denoted as \(\sin^{-1}(x)\), is the function that "undoes" the sine function. In other words, if \(\sin(x) = y\), then \(\sin^{-1}(y) = x\). Note that the range of the inverse sine function is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
02

Recall the property of the sine function in relation to negative angles

Recall that the sine function has the property that \(\sin(-x) = -\sin(x)\). In other words, the sine function is an odd function. This means that the sine function is symmetric with respect to the origin, which has important implications for the inverse sine function.
03

Understand symmetry for the inverse sine function

Given that the sine function is odd and symmetric with respect to the origin, we know that for any angle \(x\) in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), we have that \(\sin^{-1}(-\sin(x)) = -\sin^{-1}(\sin(x))\).
04

Apply the identity to the given expression

We want to prove that: $$\sin ^{-1} y+\sin ^{-1}(-y)=0$$ Let \(x = \sin^{-1}(y)\), so \(\sin(x) = y\). We know from the properties of the sine function that \(\sin(-x) = -\sin(x) = -y\). Now, we use the inverse sine function on both sides of the equation: $$-\sin^{-1}(\sin(x)) = \sin^{-1}(-y)$$ Since \(x = \sin^{-1}(y)\), we have: $$-\sin^{-1}(y) = \sin^{-1}(-y)$$
05

Combine the expressions

Finally, we can substitute this expression back into the original problem: $$\sin^{-1}(y) + \sin^{-1}(-y) = 0$$ Since we found that \(-\sin^{-1}(y) = \sin^{-1}(-y)\), the expression becomes: $$\sin^{-1}(y) - \sin^{-1}(y) = 0$$ Which simplifies to: $$0 = 0$$ This confirms the given identity: $$\sin ^{-1} y+\sin ^{-1}(-y)=0$$

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!

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

Pole in a corner A pole of length \(L\) is carried horizontally around a corner where a 3 -ft-wide hallway meets a 4 -ft-wide hallway. For \(0<\theta<\pi / 2,\) find the relationship between \(L\) and \(\theta\) at the moment when the pole simultaneously touches both walls and the corner \(P .\) Estimate \(\theta\) when \(L=10 \mathrm{ft}.\)

Daylight function for \(40^{\circ}\) N Verify that the function $$ D(t)=2.8 \sin \left(\frac{2 \pi}{365}(t-81)\right)+12 $$ has the following properties, where \(t\) is measured in days and \(D\) is the number of hours between sunrise and sunset. a. It has a period of 365 days. b. Its maximum and minimum values are 14.8 and \(9.2,\) respectively, which occur approximately at \(t=172\) and \(t=355\) respectively (corresponding to the solstices). c. \(D(81)=12\) and \(D(264) \approx 12\) (corresponding to the equinoxes).

Consider the quartic polynomial \(y=f(x)=x^{4}-x^{2}\) a. Graph \(f\) and estimate the largest intervals on which it is oneto-one. The goal is to find the inverse function on each of these intervals. b. Make the substitution \(u=x^{2}\) to solve the equation \(y=f(x)\) for \(x\) in terms of \(y .\) Be sure you have included all possible solutions. c. Write each inverse function in the form \(y=f^{-1}(x)\) for each of the intervals found in part (a).

Use the following steps to prove that \(\log _{b} x^{z}=z \log _{b} x\) a. Let \(x=b^{p}\). Solve this expression for \(p\) b. Use property E3 for exponents to express \(x^{z}\) in terms of \(b\) and \(p\) c. Compute \(\log _{b} x^{z}\) and simplify.

Make a sketch of the given pairs of functions. Be sure to draw the graphs accurately relative to each other. $$y=x^{4} \text { and } y=x^{6}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.