/*! 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 21 Evaluate each expression without... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 21

Evaluate each expression without using a calculator. (Hint: See Example 3.) (a) \(\sin \left(\arctan \frac{3}{4}\right)\) (b) \(\sec \left(\arcsin \frac{4}{5}\right)\)

Short Answer

Expert verified
(a) \(\sin \left(\arctan \frac{3}{4}\right) = \frac{3}{5} \) and (b) \(\sec (\arcsin \frac{4}{5})= \frac{5}{3} \)

Step by step solution

01

Evaluate \(\sin \left(\arctan \frac{3}{4}\right)\)

Given \(\arctan \frac{3}{4}\) we know opposite side length is 3 and adjacent side length is 4 in a right triangle. Due to Pythagorean theorem, the hypotenuse can be calculated as: \( \sqrt{3^2 + 4^2} = 5 \). Consequently, \( \sin \theta = \frac{opposite}{hypotenuse} = \frac{3}{5} \). So, \(\sin \left(\arctan \frac{3}{4}\right)=\frac{3}{5} \).
02

Evaluate \(\sec (\arcsin \frac{4}{5})\)

Here, the argument of the trigonometric function is \(\arcsin \frac{4}{5}\). In a right triangle, this implies the hypotenuse is 5 and the side opposite of the angle is 4. Using the Pythagorean theorem again, the remaining side adjacent to the angle is \( \sqrt{5^2-4^2} = 3 \). Consequently, the secant is the reciprocal of the cosine function, or \( \frac{hypotenuse}{adjacent} = \frac{5}{3} \). Hence, \(\sec (\arcsin \frac{4}{5})= \frac{5}{3} \).

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 \(P .\)

Use a graphing utility to graph \(f(x)=\sin x\) and \(g(x)=\arcsin (\sin x)\). (a) Why isn't the graph of \(g\) the line \(y=x ?\) (b) Determine the extrema of \(g\)

In Exercises 87鈥�90, solve the differential equation. $$ \frac{d y}{d x}=\frac{1}{(x-1) \sqrt{-4 x^{2}+8 x-1}} $$

Exercises 75鈥�82, find the indefinite integral using the formulas from Theorem 5.20. $$ \int \frac{1}{\sqrt{x} \sqrt{1+x}} d x $$

Find the derivative of the function. \(g(x)=3 \arccos \frac{x}{2}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.