Problem 910 Find \(\sin (\operatorname{Arcta... [FREE SOLUTION]

Chapter 28: Problem 910

Find \(\sin (\operatorname{Arctan} \mathrm{x})\), where \(\mathrm{x}\) may be any real number.

Short Answer

Expert verified

We begin with the relationship \(\alpha = \operatorname{Arctan}(x)\), where \(\tan(\alpha) = x\). Using a right triangle with angle \(\alpha\), opposite side a and adjacent side b, we set \(a=xb\). Using the Pythagorean theorem, we find the hypotenuse: \(c = b\sqrt{x^2 + 1}\). Then, we calculate the sine of the angle: \(\sin(\alpha) = \frac{a}{c} = \frac{xb}{b\sqrt{x^2 + 1}}\). Finally, we simplify the expression to obtain \(\sin(\operatorname{Arctan}(x)) = \frac{x}{\sqrt{x^2 + 1}}\).

Step by step solution

Understanding the inverse tangent function

The inverse tangent function, denoted by \(\operatorname{Arctan}(x)\), is used to find the angle whose tangent is x for any given real number x. In other words, if \(\alpha = \operatorname{Arctan}(x)\), then \(\tan(\alpha) = x\). We will work with a right triangle where one angle is \(\alpha\), and its tangent, the ratio of the opposite side (a) to the adjacent side (b), is equal to x.

Setting up the right triangle

Let's denote the opposite side of the angle \(\alpha\) in the triangle as a, the adjacent side as b, and the hypotenuse as c. From the definition of the tangent function, we know that \(\tan(\alpha) = \frac{a}{b}\). Since \(\tan(\alpha) = x\), we can say that \(a = xb\). Now, we need to find the length of the hypotenuse. From the Pythagorean theorem, we obtain the equation: \[c^2 = a^2 + b^2\]

Substituting the value of a and simplifying

Replace a with \(xb\) in the Pythagorean theorem equation: \[c^2 = (xb)^2 + b^2\] Simplify the expression, and we get: \[c^2 = x^2b^2 + b^2\]

Factoring and finding the value of c

Factor the expression to find the hypotenuse length: \[c^2 = b^2(x^2 + 1)\] Now, take the square root of both sides: \[c = b\sqrt{x^2 + 1}\]

Finding the value of sin(伪)

We want to find the value of \(\sin(\alpha)\). Since we know the opposite side (a) and the hypotenuse (c), we can use the sine formula, which is the ratio of the opposite side to the hypotenuse: \[\sin(\alpha) = \frac{a}{c}\] Replace a with \(xb\) and c with \(b\sqrt{x^2 + 1}\): \[\sin(\alpha) = \frac{xb}{b\sqrt{x^2 + 1}}\]

Simplify the expression

Simplify the expression to get the final answer: \[\sin(\alpha) = \frac{x}{\sqrt{x^2 + 1}}\] So, \(\sin(\operatorname{Arctan}(x)) = \frac{x}{\sqrt{x^2 + 1}}\).

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find \(\sin (\operatorname{Arctan} \mathrm{x})\), where \(\mathrm{x}\) may be any real number.

Short Answer

Step by step solution

Understanding the inverse tangent function

Setting up the right triangle

Substituting the value of a and simplifying

Factoring and finding the value of c

Finding the value of sin(伪)

Simplify the expression

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Logic and Functions

Applied Mathematics

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.