/*! 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 62 Draw a right triangle to simplif... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 62

Draw a right triangle to simplify the given expressions. Assume \(x>0\) $$\cos \left(\sin ^{-1}(x / 3)\right)$$

Short Answer

Expert verified
Answer: \(\frac{\sqrt{9 - x^2}}{3}\)

Step by step solution

01

Draw a right triangle

Draw a right triangle, let's say ABC, with angle A being the desired angle such that \(\sin(A) = (x/3)\). Let side BC be opposite to angle A and AB be adjacent to angle A. Then, by definition of sine, we have \(\sin(A) = \frac{BC}{AC}\). Since \(\sin(A)=\frac{x}{3}\), we can assume the length \(BC=x\) and \(AC=3\).
02

Use the Pythagorean theorem to find the third side

By the Pythagorean theorem, we have: $$AB^2 + BC^2 = AC^2.$$ Plugging in the known lengths, we get: $$AB^2 + x^2 = 3^2.$$ Now, solve for \(AB\): $$AB = \sqrt{9 - x^2}.$$
03

Find the cosine of angle A

Using the definition of cosine, we have: $$\cos(A) = \frac{AB}{AC} = \frac{\sqrt{9 - x^2}}{3}.$$ So, the simplified expression for \(\cos \left(\sin ^{-1}(x / 3)\right)\) is: $$\cos \left(\sin ^{-1}(x / 3)\right) = \frac{\sqrt{9 - x^2}}{3}.$$

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

Beginning with the graphs of \(y=\sin x\) or \(y=\cos x,\) use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility only to check your work. $$p(x)=3 \sin (2 x-\pi / 3)+1$$

Modify Exercise 84 and use property \(\mathrm{E} 2\) for exponents to prove that \(\log _{b}(x / y)=\log _{b} x-\log _{b} y\).

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. \(\sin (a+b)=\sin a+\sin b\) b. The equation \(\cos \theta=2\) has multiple real solutions. c. The equation \(\sin \theta=\frac{1}{2}\) has exactly one solution. d. The function \(\sin (\pi x / 12)\) has a period of 12 e. Of the six basic trigonometric functions, only tangent and cotangent have a range of \((-\infty, \infty)\) f. \(\frac{\sin ^{-1} x}{\cos ^{-1} x}=\tan ^{-1} x\) g. \(\cos ^{-1}(\cos (15 \pi / 16))=15 \pi / 16\) h. \(\sin ^{-1} x=1 / \sin x\)

Population model A culture of bacteria has a population of 150 cells when it is first observed. The population doubles every 12 hr, which means its population is governed by the function \(p(t)=150 \times 2^{t / 12},\) where \(t\) is the number of hours after the first observation. a. Verify that \(p(0)=150,\) as claimed. b. Show that the population doubles every \(12 \mathrm{hr}\), as claimed. c. What is the population 4 days after the first observation? d. How long does it take the population to triple in size? e. How long does it take the population to reach \(10,000 ?\)

Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture. $$\tan ^{-1} x ; \frac{\pi}{2}-\cot ^{-1} x$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision Maths

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.