/*! 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 76 Prove the following identities a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 76

Prove the following identities and give the values of \(x\) for which they are true. $$\sin \left(2 \sin ^{-1} x\right)=2 x \sqrt{1-x^{2}}$$

Short Answer

Expert verified
Answer: The identity holds true for all \(x\in[-1,1]\), as that is the domain of the arcsin function.

Step by step solution

01

Apply the double angle formula

The formula for the sine of twice an angle is given by: $$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$$ Let's apply this formula to our given identity, replacing \(\theta\) with \(\sin^{-1}(x)\): $$\sin \left(2 \sin ^{-1} x\right) = 2\sin(\sin^{-1}(x))\cos(\sin^{-1}(x))$$
02

Use the properties of inverse functions

The properties of inverse functions tell us that \(\sin(\sin^{-1}(x)) = x\) as long as x is in the domain of \(\sin^{-1}(x)\). The domain of \(\sin^{-1}(x)\) is \([-1,1]\), so as long as \(x\in[-1,1]\), we can make this substitution: $$2\sin(\sin^{-1}(x))\cos(\sin^{-1}(x)) = 2x\cos(\sin^{-1}(x))$$
03

Find \(\cos(\sin^{-1}(x))\) based on the Pythagorean identity

Recall the Pythagorean identity for sine and cosine: $$\sin^2(\theta) + \cos^2(\theta) = 1$$ With our angle \(\theta = \sin^{-1}(x)\), we have: $$\sin^2(\sin^{-1}(x)) + \cos^2(\sin^{-1}(x)) = 1$$ Substituting \(\sin(\sin^{-1}(x)) = x\), we get: $$x^2 + \cos^2(\sin^{-1}(x)) = 1$$ Now, we can solve for \(\cos(\sin^{-1}(x))\): $$\cos^2(\sin^{-1}(x)) = 1 - x^2$$ $$\cos(\sin^{-1}(x)) = \sqrt{1-x^2}$$
04

Substitute the expression of \(\cos(\sin^{-1}(x))\) into our equation

Now we can substitute the expression for \(\cos(\sin^{-1}(x))\) back into the equation we derived earlier: $$2x\cos(\sin^{-1}(x)) = 2x\sqrt{1-x^2}$$
05

Conclusion and values of x

We've shown that the given identity is true: $$\sin \left(2 \sin ^{-1} x\right)=2 x \sqrt{1-x^{2}}$$ This identity holds true for all \(x\in[-1,1]\), as that is the domain of the arcsin function.

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

Witch of Agnesi Let \(y\left(x^{2}+4\right)=8\) (see figure). a. Use implicit differentiation to find \(\frac{d y}{d x}\) b. Find equations of all lines tangent to the curve \(y\left(x^{2}+4\right)=8\) when \(y=1\) c. Solve the equation \(y\left(x^{2}+4\right)=8\) for \(y\) to find an explicit expression for \(y\) and then calculate \(\frac{d y}{d x}\) d. Verify that the results of parts (a) and (c) are consistent.

The following limits equal the derivative of a function \(f\) at a point a. a. Find one possible \(f\) and \(a\) b. Evaluate the limit. $$\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-\frac{1}{2}}{h}$$

Prove the following identities and give the values of \(x\) for which they are true. $$\cos \left(2 \sin ^{-1} x\right)=1-2 x^{2}$$

A hot-air balloon is \(150 \mathrm{ft}\) above the ground when a motorcycle passes directly beneath it (traveling in a straight line on a horizontal road) going \(40 \mathrm{mi} / \mathrm{hr}(58.67 \mathrm{ft} / \mathrm{s})\) If the balloon is rising vertically at a rate of \(10 \mathrm{ft} / \mathrm{s},\) what is the rate of change of the distance between the motorcycle and the balloon 10 seconds later?

Use the following table to find the given derivatives. $$\begin{array}{llllll} x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 5 & 4 & 3 & 2 & 1 \\ f^{\prime}(x) & 3 & 5 & 2 & 1 & 4 \\ g(x) & 4 & 2 & 5 & 3 & 1 \\ g^{\prime}(x) & 2 & 4 & 3 & 1 & 5 \end{array}$$ $$\left.\frac{d}{d x}\left[\frac{x f(x)}{g(x)}\right]\right|_{x=4}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete 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.