Right Triangle Trigonometry

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iKidney Failure CHOOSING A T R E ATMENT THAT ’ SR I G H TF O RYO U National Institutes of HealthNational Institute of Diabetes and Digestive and Kidney DiseasesKidney Failure CHOOSING A T R E ATMENT THAT ’ SR I G H TF O RYO U C o n t e n t sIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1When Your Kidneys Fail . . . . . . . . . . . . . . . . . . . . . . . . . . 1Treatment Choice: Hemodialysis . . . . . . . . . . . . . . . . . . . 2Treatment Choice: Peritoneal Dialysis . . . . . . . . . . . . . . . 9Treatment Choice: Kidney Transplantation . . . . . . . . . . . 1 5Treatment Choice: Refusing or Withdrawing From Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2Paying for Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 1I n t ro d u c t i o nYour kidneys filter wastes from your blood and regulate otherfunctions of your body. When your kidneys fail, you needtreatment to replace the work of healthy kidneys to survive.Developing kidney failure means that you have some decisionsto make about your treatment. If you choose to receive treat-ment, your choices are hemodialysis, peritoneal dialysis, andkidney transplantation. Each of them has advantages and dis-advantages. You may also choose to forgo treatment. Bylearning about your choices, you can work with your doctorto decide what’s best for you. No matter which treatmentyou choose, you’ll need to make some changes in your life,including how you eat and plan your activities. But with thehelp of your health care team, family, and friends, you canlead a full, active life.When Your Kidneys Fa i lHealthy kidneys clean your blood by removing excess fluid,minerals, and wastes. They also make hormones that keepyour bones strong and your blood healthy. When your kid-neys fail, harmful wastes build up in your body, your bloodpressure may rise, and your body may retain excess fluid andnot make enough red blood cells. When this happens, youneed treatment to replace the work of your failed kidneys. 2Treatment Choice: HemodialysisP u r p o s eHemodialysis cleans and filters your blood using a machine totemporarily rid your body of harmful wastes, extra salt, andextra water. Hemodialysis helps control blood pressure andhelps your body keep the proper balance of important chemi-cals such as potassium, sodium, calcium, and bicarbonate.How It Wo r k sHemodialysis uses a special filter called a dialyzer that func-tions as an artificial kidney to clean your blood. During treat-ment, your blood travels through tubes into the dialyzer,which filters out wastes and extra water. Then the cleanedblood flows through another set of tubes back into your body.The dialyzer is connected to a machine that monitors bloodflow and removes wastes from the blood.H e m o d i a l y s i s .Heparin pump(to preventc l o t t i n g )Dialyzer inflowpressure monitorD i a l y z e rA r t e r i a lpressure monitorBlood pumpBlood removedfor cleansingClean bloodreturned tob o d yAir detectorc l a m pAir trap andair detectorVenous pressure monitor 3Hemodialysis is usually needed three times a week. Eachtreatment lasts from 3 to 5 or more hours. During treatment,you can read, write, sleep, talk, or watch TV. Getting ReadyIf you choose hemodialysis, several months before your firsttreatment, an access to your bloodstream will need to becreated. You may need to stay overnight in the hospital, butmany patients have their access placed on an outpatient basis.This access provides an Right Triangle Trigonometry Right Triangle Trigonometry By: OpenStaxCollege We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle: cos t = x sin t = y In this section, we will see another way to define trigonometric functions using properties of right triangles Using Right Triangles to Evaluate Trigonometric Functions In earlier sections, we used a unit circle to define the trigonometric functions In this section, we will extend those definitions so that we can apply them to right triangles The value of the sine or cosine function of t is its value at t radians First, we need to create our right triangle [link] shows a point on a unit circle of radius If we drop a vertical line segment from the point (x, y) to the x-axis, we have a right triangle whose vertical side has length y and whose horizontal side has length x We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle We know 1/34 Right Triangle Trigonometry cos t = x =x Likewise, we know sin t = y =y These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using (x, y) coordinates To be able to use these ratios freely, we will give the sides more general names: Instead of x, we will call the side between the given angle and the right angle the adjacent side to angle t (Adjacent means “next to.”) Instead of y, we will call the side most distant from the given angle the opposite side from anglet And instead of 1, we will call the side of a right triangle opposite the right angle the hypotenuse These sides are labeled in [link] The sides of a right triangle in relation to angle t Understanding Right Triangle Relationships Given a right triangle with an acute angle of t, sin(t) = opposite hypotenuse cos(t) = adjacent hypotenuse tan(t) = opposite adjacent A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.” How To 2/34 Right Triangle Trigonometry Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle Find the sine as the ratio of the opposite side to the hypotenuse Find the cosine as the ratio of the adjacent side to the hypotenuse Find the tangent is the ratio of the opposite side to the adjacent side Evaluating a Trigonometric Function of a Right Triangle Given the triangle shown in [link], find the value of cos α The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so: cos(α) = adjacent hypotenuse = 15 17 Try It Given the triangle shown in [link], find the value of sin t 25 3/34 Right Triangle Trigonometry Relating Angles and Their Functions When working with right triangles, the same rules apply regardless of the orientation of the triangle In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in [link] The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa The side adjacent to one angle is opposite the other We will be asked to find all six trigonometric functions for a given angle in a triangle Our strategy is to find the sine, cosine, and tangent of the angles first Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent How To Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles If needed, draw the right triangle and label the angle provided Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle Find the required function: ◦ sine as the ratio of the opposite side to the hypotenuse ◦ cosine as the ratio of the adjacent side to the hypotenuse ◦ tangent as the ratio of the opposite side to the adjacent side ◦ secant as the ratio of the hypotenuse to the adjacent side ◦ cosecant as the ratio of the hypotenuse to the opposite side ◦ cotangent as the ratio of the adjacent side to the opposite side Evaluating Trigonometric Functions of Angles Not in Standard Position Using the triangle shown in [link], evaluate sin α, cos α, tan α, sec α, csc α, and cot α 4/34 Right Triangle Trigonometry sin α = opposite α = hypotenuse cos α = adjacent to α = hypotenuse tan α = opposite α = adjacent to α sec α = hypotenuse = adjacent to α csc α = hypotenuse = opposite α cot α = adjacent to α = opposite α Try It Using the triangle shown in [link], evaluate sin t, cos t, tan t, sec t, csc t, and cot t sin t = sec t = 33 65 , 65 56 , cos t = csc t = 56 65 , 65 33 , tan t = 33 56 , cot t = 56 33 5/34 ... BETTERWRITINGRIGHT NOW!Using Words to Your AdvantageNEW YORKFrancine D. Galko Copyright © 2001 LearningExpress, LLC.All rights reserved under International and Pan-American Copyright Conventions.Published in the United States by LearningExpress, LLC, New York.Library of Congress Cataloging-in-Publication Data:Galko, Francine.Better writing right now : using words to your advantage / by Francine Galko.—1st ed.p. cm.ISBN 1-57685-402-71. English language—Rhetoric. 2. Report writing. 3. Business writing. I. Title.PE1408 .G25 2002808'.042—dc21 2001050784ISBN 1-57685-402-7Printed in the United States of America987654321First EditionFor more information or to place an order, contact LearningExpress at:900 BroadwaySuite 604New York, NY 10003Or visit us at:www.learnatest.com ABOUT THE AUTHORFrancine D. Galko is currently a freelance writer, editor, and project manager. She has edited pre-GED andGED math preparation work texts, and has also written a basic math and algebra study guide with practicematerials and interactive CD-ROMs. In addition, Ms. Galko has written and edited other science, languagearts, ESL, EFL, and instructional materials. She currently resides in Dallas, Texas. ContentsIntroduction ixSection 1: Deciding What to Say—Preparing to Write 1Lesson 1: Getting Started 3Lesson 2: Choosing Your Own Topic 11Lesson 3: Using Prewriting Strategies 19Lesson 4: Organizing Your Ideas and Outlining Your Paper 33Section 2: Start Writing!—The Drafting Process 45Lesson 5: Starting to Draft Your Paper/Drafting Your Paper 47Lesson 6: Convincing Your Reader 55Lesson 7: Beginning and Ending Your Paper 63Section 3: Evaluating What You’ve Written—Revising and Editing 69Lesson 8: Revising Your Paper 71Lesson 9: Checking the Focus and Organization of Your Paper 77Lesson 10: Editing Your Paper 85Lesson 11: Being Clear and Concise 101Lesson 12: Writing with Style 109Section 4: Special Writing Situations 115Lesson 13: Essay Exams 117Lesson 14: Research Papers 123Section 5: Writing for the Workplace 135Lesson 15: Business Writing 137Lesson 16: Resumes and Cover Letters 139Lesson 17: Writing Business Letters 157Lesson 18: Writing Memos and Emails 167Lesson 19: Writing Reports 177Appendix: Model Essays and Workplace Writing 183Answers 215BETTER WRITING RIGHT NOW!v IntroductionLet’s say you’re at the bookstore and you’re trying todecide whether or not to buy this book. You wonder:Will it really help me write better? Is it any different fromthe other books on the shelf? How can this bookimprove my writing? If these are some of the ques-tions you have, then read on—you’ll find the answershere!etter Writing Right Now is a step-by-step guide to writing. It takes you from the blank page andwalks you through the steps of the writing process so that you can conquer any school writingassignment—including timed essay exams and research papers. It also provides tips and formats you can usefor writing resumes, cover letters, general business letters, memos, e-mails, and reports for work. Along theway, you’ll learn basic writing skills, and you’ll gain the confidence you need to succeed in any situation thatrequires you to write.This book gives you more than the information you need to become a better writer. It also gives youexample after example of strategies that work and provides opportunities to practice those strategies. Takeadvantage of each practice, because here you can safely experiment with techniques and develop expert skillsbefore you have to use them for class, work, or correspondence. Your work in this book can be for your eyesonly—so stretch your fingers, stretch your imagination, and don’t be afraid to see your writing take shape.ISTHISBOOK FORYOU?This DIGITAL RIGHT MANAGEMENTNhóm: LION1. Huỳnh Trọng Khiêm2. Phạm Ngọc Khanh3. Nguyễn Thành Danh4. Nguyễn Văn Lợi5. Võ Nguyễn Bảo Tịnh NỘI DUNG1. Giới Thiệu Digital Right Management 2. Các Công Cụ Bảo Mật trong DRM3. Các Mô Hình Trong DRM4. Khả Năng Bị Tấn Công Của DRM 1. GIỚI THIỆU DRMDigital rights management (DRM) ra đời nhằm kiểm soát và bảo vệ bản quyền các phương tiện truyền thông điện tử gồm âm nhạc kỹ thuật số và phim ảnh, cũng như các dữ liệu khác được lưu trữ và chuyển giao kỹ thuật số.DRM giúp cho các nhà xuất bản các phương tiện truyền thông chắn chắn rằng các nội dung kỹ thuật số chỉ được sử dụng bởi những người đã trả tiền cho nó. 2. 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[...]... hỗ trợ nhiều giao thức ở bên trên nó không riêng HTTP mà còn có thể là FTP, NNTP NGÔN NGỮ MÔ TẢ BẢN QUYỀN  Ngôn ngữ mô tả quyền (Rights Expression Language-REL)  Các ngôn ngữ mô tả quyền :  Phổ biến hiện nay là ODRL (Open Digital Rights Language)  XrML (eXtensible rights Makup Language)  MPEG (Moving Pictures Expert Group) NGÔN NGỮ MÔ TẢ BẢN QUYỀN (tt)  Một số yêu cầu của REL :  REL phải đủ... tiết thanh toán, thông tin bảo mật, các chi tiết kỹ thuật xử lý cũng như luồng dữ liệu NGÔN NGỮ MÔ TẢ BẢN QUYỀN (tt)  Hai nhân tố cơ bản của REL :  Khái niệm ngôn ngữ quyền (rights language concept)  Từ điển dữ liệu quyền (Rights Data Dictionary viết tắt là RDD) MÔ HÌNH DRM  Hệ thống DRM là một môi trường đáng tin cậy cho việc đảm bảo các nội dung kỹ thuật số giữa những đối tác  Các chức năng45-45-90 Right Triangles A right triangle with two angles each measuring 45° is called an isosceles right triangle. In an isosceles right triangle: ■ The length of the hypotenuse is ͙2 ෆ multiplied by the length of one of the legs of the triangle. ■ The length of each leg is multiplied by the length of the hypotenuse. x = y = × ᎏ 1 1 0 ᎏ = = 5͙2 ෆ 30-60-90 Triangles In a right triangle with the other angles measuring 30° and 60°: ■ The leg opposite the 30-degree angle is half the length of the hypotenuse. (And, therefore, the hypotenuse is two times the length of the leg opposite the 30-degree angle.) ■ The leg opposite the 60 degree angle is ͙3 ෆ times the length of the other leg. Example: x = 2 × 7 = 14 and y = 7͙3 ෆ 60° 30° x y 7 60° 30° 2s s s √ ¯¯¯ 3 10͙2 ෆ ᎏ 2 ͙2 ෆ ᎏ 2 10 x y ͙2 ෆ ᎏ 2 45° 45° –THE SAT MATH SECTION– 133 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 133 Triangle Trigonometry There are special ratios we can use with right triangles. They are based on the trigonometric functions called sine, cosine, and tangent. The popular mnemonic to use is: SOH CAH TOA For an angle, θ, within a right triangle, we can use these formulas: sin θ = cos θ = tan θ = TRIG VALUES OF SOME COMMON ANGLES sin cos tan 30° ᎏ 1 2 ᎏ 45° 1 60° ᎏ 1 2 ᎏ ͙3 ෆ Whereas it is possible to solve some right triangle questions using the knowledge of 30-60-90 and 45-45- 90 triangles, an alternative method is to use trigonometry. For example, solve for x below. Using the knowledge that cos 60° = ᎏ 1 2 ᎏ , just sub- stitute into the equation: ᎏ 5 x ᎏ = ᎏ 1 2 ᎏ , so x = 10. Circles A circle is a closed figure in which each point of the cir- cle is the same distance from a fixed point called the center of the circle. Angles and Arcs of a Circle ■ An arc is a curved section of a circle. A minor arc is smaller than a semicircle and a major arc is larger than a semicircle. ■ A central angle of a circle is an angle that has its vertex at the center and that has sides that are radii. ■ Central angles have the same degree measure as the arc it forms. Length of an Arc To find the length of an arc, multiply the circumference of the circle, 2πr,where r = the radius of the circle by the fraction ᎏ 36 x 0 ᎏ , with x being the degree measure of the arc or central angle of the arc. Example: Find the length of the arc if x = 36 and r = 70. L = ᎏ 3 3 6 6 0 ᎏ × 2(π)70 L = ᎏ 1 1 0 ᎏ × 140π L = 14π r x r o M i n o r A r c M a j o r A r c Central Angle 60 o 5 x ͙3 ෆ ᎏ 2 ͙2 ෆ ᎏ 2 ͙2 ෆ ᎏ 2 ͙3 ෆ ᎏ 3 ͙3 ෆ ᎏ 2 opposite hypotenuse adjacent hypotenuse opposite adjacent To find sin To find cos To find tan Opposite ᎏ Adjacent Adjacent ᎏᎏ Hypotenuse Opposite ᎏᎏ Hypotenuse –THE SAT MATH SECTION– 134 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 134 Area of a Sector The area of a sector is found in a similar way. To find the area of a sector, simply multiply the area of a circle (π)r 2 by the fraction ᎏ 36 x 0 ᎏ , again using x as the degree measure of the central angle. Example: Given x = 60 and r = 8, find the area of the sector. A = ᎏ 3 6 6 0 0 ᎏ × (π)8 2 A = ᎏ 1 6 ᎏ × 64(π) A = ᎏ 6 6 4 ᎏ (π) A = ᎏ 3 3 2 ᎏ (π) Polygons and Parallelograms A polygon is a figure with three or more sides. Terms Related to ... 14/34 Right Triangle Trigonometry • Right- triangle trigonometry permits the measurement of inaccessible heights and distances • The unknown height or distance can be found by creating a right triangle. .. Trigonometry Using Right Triangle Trigonometry to Solve Applied Problems Right- triangle trigonometry has many practical applications For example, the ability to compute the lengths of sides of a triangle. .. Finding Trig Functions Using a Right Triangle Relate Trig Functions to Sides of a Right Triangle Determine Six Trig Functions from a Triangle Determine Length of Right Triangle Side Visit this website