SOLVED:Compute \sinh (\ln 5) And \tanh (3 \ln 5) Without Using A ...

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Question

Compute $\sinh (\ln 5)$ and $\tanh (3 \ln 5)$ without using a calculator. Compute $\sinh (\ln 5)$ and $\tanh (3 \ln 5)$ without using a calculator. Calculus for AP Jon Rogawski & Ray… 2nd Edition Chapter 1, Problem 36 View All Chapters

Step 1

Step 1: We start by using the definition of the hyperbolic sine function: \[\sinh(x) = \frac{e^x - e^{-x}}{2}\] So, we have: \[\sinh(\ln 5) = \frac{e^{\ln 5} - e^{-\ln 5}}{2}\] Show more…

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Key Concepts

Exponential and Logarithmic Identities Exponential and logarithmic identities are core tools in simplifying expressions that involve exponential and logarithm functions. A key identity is that e^(ln x) equals x, and more generally, e^(a ln x) equals x^a. This relationship is fundamental to transforming expressions like sinh(ln x) or tanh(3 ln x) into forms that can be easily simplified, showing the interplay between exponentiation and logarithms. Hyperbolic Functions Hyperbolic functions, including sinh and tanh, are analogous to the trigonometric functions but are defined using exponential functions. They are essential in various branches of mathematics and physics, particularly in solving differential equations and describing hyperbolic geometry. Understanding their definitions—involving combinations of exponential functions—allows one to simplify and compute their values without numerical computation.

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Transcript

00:01 So we will solve the following sign of national log of 5 and tangent of 3, natural log of 5 without using a calculator through the method of using these following hyperbolic functions as seen in the textbook. 00:18 And to calculate what the hyperbolic function would be for tangent, we know that tangent is simply equal to the sign over cosine. 00:27 So that's exactly what i did. 00:29 I divided this by this to get this following equation. 00:37 So let's work on the first half of this problem, sign of a national log of 5. 00:42 What i did was i treated the natural log of 5 as x, and i plugged that into the hyperbolic function for a sign to get this following equation. 00:53 Sign of the national log of 5 equals e to the national of 5 minus, minus e to the natural log, or to the negative natural log of 5 all over 2. 01:06 And when i look at this portion of the problem, i know that this is the same exact value as negative e to the natural log of 1 5th. 01:27 And now that we've clarified that, we can simplify it even further, because when we have the e, to the natural log of something, it will simplify to whatever is in the parentheses for the natural log. 01:44 So this formula will turn into 5 minus 1 5th, all over 2. 01:58 And we can turn this into 25 over 5 for the least common denominator, one fifth while over two. 02:17 So it's 24 over five divided by two. 02:28 So we get our answer as 24 over 10. 02:39 Great. 02:41 Let's use the same principles on our tangent function... Need help? Use Ace Ace is your personal tutor. It breaks down any question with clear steps so you can learn. Start Using Ace Ace is your personal tutor for learning Step-by-step explanations Instant summaries Summarize YouTube videos Understand textbook images or PDFs Study tools like quizzes and flashcards Listen to your notes as a podcast

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