/*! 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 22 Evaluate each expression without... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 22

Evaluate each expression without using a calculator. (Hint: See Example 3.) (a) \(\tan \left(\arccos \frac{\sqrt{2}}{2}\right)\) (b) \(\cos \left(\arcsin \frac{5}{13}\right)\)

Short Answer

Expert verified
(a) The value of the expression \(\tan \left(\arccos \frac{\sqrt{2}}{2}\right)\) is 1. \n(b) The value of the expression \(\cos \left(\arcsin \frac{5}{13}\right)\) is \(\frac{12}{13}\).

Step by step solution

01

Evaluate Expression (a)

Expression (a) is \(\tan \left(\arccos \frac{\sqrt{2}}{2}\right)\). It can be evaluated by realising that \(\arccos \frac{\sqrt{2}}{2}\) is equal to \(\frac{\pi}{4}\) because \(\cos( \frac{\pi}{4}) = \frac{\sqrt{2}}{2}\). This implies that the expression can be simplified to \(\tan(\frac{\pi}{4})\), because of the inverse cosine transformation. The tangent of \(\frac{\pi}{4}\) equals 1.
02

Evaluate Expression (b)

Expression (b) is \(\cos \left(\arcsin \frac{5}{13}\right)\). It can be evaluated by using the Pythagorean trigonometric identity: \( \sin^2(x) + \cos^2(x) = 1 \). Given that \(\sin(x) = \frac{5}{13}\), we can solve for \(\cos(x)\) to obtain \(\cos(x) = \sqrt{1 - \sin^2(x)} = \sqrt{1-\left(\frac{5}{13}\right)^2} = \sqrt{1-\frac{25}{169}} = \frac{12}{13}\). So the value of the expression is \(\cos \left(\arcsin \frac{5}{13}\right) = \frac{12}{13}\). Note here that we assumed the angle to be in the first quadrant, hence positive cosine value.

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

From the vertex \((0, c)\) of the catenary \(y=c \cosh (x / c)\) a line \(L\) is drawn perpendicular to the tangent to the catenary at point \(P .\) Prove that the length of \(L\) intercepted by the axes is equal to the ordinate \(y\) of the point \(P .\)

Area In Exercises \(125-128\) , find the area of the region bounded by the graphs of the equations. Use a graphing utility to graph the region and verify your result. $$ y=x e^{-x^{2} / 4}, y=0, x=0, x=\sqrt{6} $$

Analyze and sketch a graph of the function. Identify any relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. \(f(x)=\arcsin (x-1)\)

Deriving an Inequality Given \(e^{x} \geq 1\) for \(x \geq 0,\) it follows that $$ \int_{0}^{x} e^{t} d t \geq \int_{0}^{x} 1 d t $$ Perform this integration to derive the inequality $$ \begin{array}{l}{e^{x} \geq 1+x} \\ {\text { for } x \geq 0}\end{array} $$

Find an equation of the tangent line to the graph of the function at the given point. \(y=4 x \arccos (x-1), \quad(1,2 \pi)\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

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.