Right Triangle Trigonometry Solving Word Problems Answer Key

Advertisement

Right Triangle Trigonometry: Solving Word Problems – Answer Key & Strategies



Are you wrestling with right triangle trigonometry word problems? Do you stare blankly at diagrams, unsure where to even begin? You're not alone! Many students find these problems challenging, but with the right approach and a solid understanding of the underlying principles, they become manageable—even enjoyable! This comprehensive guide provides you not only with sample problems and their solutions (your very own answer key!) but also with strategies to conquer any right triangle trigonometry word problem you encounter. We’ll break down the process step-by-step, equipping you with the tools to succeed.

Understanding the Fundamentals: SOH CAH TOA



Before diving into word problems, let's refresh our understanding of the fundamental trigonometric ratios: sine, cosine, and tangent. Remember the mnemonic SOH CAH TOA:

SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent

These ratios relate the angles and sides of a right-angled triangle. Understanding which side is opposite, adjacent, and the hypotenuse relative to the angle you're working with is crucial.


Step-by-Step Guide to Solving Right Triangle Trigonometry Word Problems



Tackling word problems requires a systematic approach. Here's a breakdown of the steps involved:

1. Draw a Diagram: This is the most critical step. Visualizing the problem with a well-labeled right-angled triangle clarifies the relationships between the given information and the unknown quantities. Label the sides (opposite, adjacent, hypotenuse) relative to the known or unknown angle.

2. Identify the Known and Unknown: Carefully read the problem to identify what information is given (angles, side lengths) and what you need to find.

3. Choose the Correct Trigonometric Ratio: Based on the known and unknown quantities, select the appropriate trigonometric ratio (sine, cosine, or tangent) that relates them.

4. Set Up and Solve the Equation: Substitute the known values into the chosen trigonometric ratio, forming an equation. Solve this equation using algebraic manipulation to find the unknown quantity. Remember to use your calculator appropriately, ensuring your calculator is in the correct angle mode (degrees or radians).

5. Check Your Answer: Does your answer make sense in the context of the problem? Is the calculated side length reasonable? A quick review can help catch errors.


Example Word Problems with Detailed Solutions (Your Answer Key!)



Let's work through a few examples to solidify your understanding.

Example 1: A ladder leans against a wall. The ladder is 10 meters long, and the base of the ladder is 3 meters from the wall. Find the angle the ladder makes with the ground.

Solution:

1. Diagram: Draw a right-angled triangle with the ladder as the hypotenuse (10m), the distance from the wall as the adjacent side (3m), and the height the ladder reaches on the wall as the opposite side (unknown).

2. Known/Unknown: Known: hypotenuse = 10m, adjacent = 3m. Unknown: angle (let's call it θ).

3. Trigonometric Ratio: We have the adjacent and hypotenuse, so we use cosine: cos θ = Adjacent / Hypotenuse

4. Equation & Solution: cos θ = 3/10. θ = cos⁻¹(3/10) ≈ 72.5°

5. Check: The angle seems reasonable given the context.


Example 2: A surveyor measures the angle of elevation to the top of a building to be 30°. The surveyor is standing 100 meters from the building. How tall is the building?

Solution: (Similar step-by-step process using tangent, leading to the solution involving the tangent ratio and resulting in the building height.)


Example 3: A kite is flying at an angle of elevation of 40° from the ground. If the string is 200 meters long, how high is the kite above the ground?

Solution: (Again following the steps, this problem also utilizes the sine ratio to find the kite's height.)


Advanced Techniques and Considerations



As you progress, you may encounter more complex problems involving multiple triangles or requiring the use of trigonometric identities. Practice is key to mastering these advanced techniques. Look for problems that challenge your understanding and push your problem-solving skills.

Conclusion



Mastering right triangle trigonometry word problems is a crucial skill in mathematics and its applications. By following the systematic approach outlined above, practicing diligently, and working through diverse examples, you can transform these initially daunting problems into opportunities to showcase your problem-solving prowess. Remember to always draw a diagram, carefully identify the known and unknown quantities, and choose the appropriate trigonometric ratio. With consistent effort, you'll confidently navigate the world of right triangle trigonometry word problems.


FAQs



1. What if I don't have a calculator? While a calculator is helpful for solving trigonometric equations, understanding the underlying concepts and using trigonometric tables (if available) are still valuable skills.

2. Can I use different trigonometric ratios to solve the same problem? Often, yes! Different ratios might lead to slightly different calculations but should yield the same answer (within rounding errors).

3. How do I handle problems with multiple triangles? Break the problem down into smaller, simpler triangles. Solve for the missing information in each triangle sequentially, using the results from one triangle to solve for the unknowns in the next.

4. What are some common mistakes to avoid? Common mistakes include incorrect labeling of sides, selecting the wrong trigonometric ratio, and calculator errors (especially regarding degrees vs. radians).

5. Where can I find more practice problems? Numerous online resources, textbooks, and practice workbooks offer extensive practice problems with varying difficulty levels. Utilize these resources to solidify your understanding and build your confidence.


  right triangle trigonometry solving word problems answer key: Pre-Calculus Workbook For Dummies Yang Kuang, Michelle Rose Gilman, Elleyne Kase, 2011-04-12 This hands-on workbook helps students master basic pre-calculus concepts and practice the types of problems they'll encounter in the course. Students will get hundreds of valuable exercises, problem-solving shortcuts, plenty of workspace, thorough explanations, and step-by-step solutions to every problem.
  right triangle trigonometry solving word problems answer key: Pre-Calculus Workbook For Dummies? Michelle Rose Gilman, Christopher Burger, Karina Neal, 2009-06-24 Get the confidence and the math skills you need to get started with calculus! Are you preparing for calculus? This easy-to-follow, hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in your cour sework. You get valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step solutions to every problem. You'll also memorize the most frequently used equations, see how to avoid common mistakes, understand tricky trig proofs, and much more. 100s of Problems! Detailed, fully worked-out solutions to problems The inside scoop on quadratic equations, graphing functions, polynomials, and more A wealth of tips and tricks for solving basic calculus problems
  right triangle trigonometry solving word problems answer key: Elementary College Geometry Henry Africk, 2004
  right triangle trigonometry solving word problems answer key: Algebra and Trigonometry Jay P. Abramson, Valeree Falduto, Rachael Gross (Mathematics teacher), David Lippman, Rick Norwood, Melonie Rasmussen, Nicholas Belloit, Jean-Marie Magnier, Harold Whipple, Christina Fernandez, 2015-02-13 The text is suitable for a typical introductory algebra course, and was developed to be used flexibly. While the breadth of topics may go beyond what an instructor would cover, the modular approach and the richness of content ensures that the book meets the needs of a variety of programs.--Page 1.
  right triangle trigonometry solving word problems answer key: ACT Math For Dummies Mark Zegarelli, 2011-06-28 Multiply your chances of success on the ACT Math Test The ACT Mathematics Test is a 60-question, 60-minute subtest designed to measure the mathematical skills students have typically acquired in courses taken by the end of 11th grade, and is generally considered to be the most challenging section of the ACT. ACT Math For Dummies is an approachable, easy-to-follow study guide specific to the Math section, complete with practice problems and strategies to help you prepare for exam day. Review chapters for algebra, geometry, and trigonometry Three practice tests modeled from questions off the most recent ACT tests Packed with tips, useful information, and strategies ACT Math For Dummies is your one-stop guide to learn, review, and practice for the test!
  right triangle trigonometry solving word problems answer key: CK-12 Calculus CK-12 Foundation, 2010-08-15 CK-12 Foundation's Single Variable Calculus FlexBook introduces high school students to the topics covered in the Calculus AB course. Topics include: Limits, Derivatives, and Integration.
  right triangle trigonometry solving word problems answer key: Precalculus Jay P. Abramson, Valeree Falduto, Rachael Gross (Mathematics teacher), David Lippman, Melonie Rasmussen, Rick Norwood, Nicholas Belloit, Jean-Marie Magnier, Harold Whipple, Christina Fernandez, 2014-10-23 Precalculus is intended for college-level precalculus students. Since precalculus courses vary from one institution to the next, we have attempted to meet the needs of as broad an audience as possible, including all of the content that might be covered in any particular course. The result is a comprehensive book that covers more ground than an instructor could likely cover in a typical one- or two-semester course; but instructors should find, almost without fail, that the topics they wish to include in their syllabus are covered in the text. Many chapters of OpenStax College Precalculus are suitable for other freshman and sophomore math courses such as College Algebra and Trigonometry; however, instructors of those courses might need to supplement or adjust the material. OpenStax will also be releasing College Algebra and Algebra and trigonometry titles tailored to the particular scope, sequence, and pedagogy of those courses.--Preface.
  right triangle trigonometry solving word problems answer key: Compiled and Solved Problems in Geometry and Trigonometry Florentin Smarandache, 2015-05-01 This book is a translation from Romanian of Probleme Compilate şi Rezolvate de Geometrie şi Trigonometrie (University of Kishinev Press, Kishinev, 169 p., 1998), and includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students.
  right triangle trigonometry solving word problems answer key: Trigonometry For Dummies Mary Jane Sterling, 2014-02-06 A plain-English guide to the basics of trig Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology. From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English and offering lots of easy-to-grasp example problems. It also explains the why of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers. Tracks to a typical Trigonometry course at the high school or college level Packed with example trig problems From the author of Trigonometry Workbook For Dummies Trigonometry For Dummies is for any student who needs an introduction to, or better understanding of, high-school to college-level trigonometry.
  right triangle trigonometry solving word problems answer key: School Library Journal , 1986
  right triangle trigonometry solving word problems answer key: Master the GED - 2011 Peterson's, 2010-07-07 Master the GED 2011 with CD is a comprehensive guide that offers the essential test-prep and review material for the high school equivalency diploma test. Includes three full-length practice exams, with detailed answer explanations for every question. Original.
  right triangle trigonometry solving word problems answer key: Master the GED 2011 (w/CD) Peterson's, 2010-07-07 This edition includes three full-length practice exams in book and access to two full-length practice exams on the accompanying CD. Original.
  right triangle trigonometry solving word problems answer key: Open Middle Math Robert Kaplinsky, 2023-10-10 This book is an amazing resource for teachers who are struggling to help students develop both procedural fluency and conceptual understanding.. --Dr. Margaret (Peg) Smith, co-author of5 Practices for Orchestrating Productive Mathematical Discussions Robert Kaplinsky, the co-creator of Open Middle math problems, brings hisnew class of tasks designed to stimulate deeper thinking and lively discussion among middle and high school students in Open Middle Math: Problems That Unlock Student Thinking, Grades 6-12. The problems are characterized by a closed beginning,- meaning all students start with the same initial problem, and a closed end,- meaning there is only one correct or optimal answer. The key is that the middle is open- in the sense that there are multiple ways to approach and ultimately solve the problem. These tasks have proven enormously popular with teachers looking to assess and deepen student understanding, build student stamina, and energize their classrooms. Professional Learning Resource for Teachers: Open Middle Math is an indispensable resource for educators interested in teaching student-centered mathematics in middle and high schools consistent with the national and state standards. Sample Problems at Each Grade: The book demonstrates the Open Middle concept with sample problems ranging from dividing fractions at 6th grade to algebra, trigonometry, and calculus. Teaching Tips for Student-Centered Math Classrooms: Kaplinsky shares guidance on choosing problems, designing your own math problems, and teaching for multiple purposes, including formative assessment, identifying misconceptions, procedural fluency, and conceptual understanding. Adaptable and Accessible Math: The tasks can be solved using various strategies at different levels of sophistication, which means all students can access the problems and participate in the conversation. Open Middle Math will help math teachers transform the 6th -12th grade classroom into an environment focused on problem solving, student dialogue, and critical thinking.
  right triangle trigonometry solving word problems answer key: How to Solve Word Problems in Calculus Eugene Don, Benay Don, 2001-07-21 Considered to be the hardest mathematical problems to solve, word problems continue to terrify students across all math disciplines. This new title in the World Problems series demystifies these difficult problems once and for all by showing even the most math-phobic readers simple, step-by-step tips and techniques. How to Solve World Problems in Calculus reviews important concepts in calculus and provides solved problems and step-by-step solutions. Once students have mastered the basic approaches to solving calculus word problems, they will confidently apply these new mathematical principles to even the most challenging advanced problems.Each chapter features an introduction to a problem type, definitions, related theorems, and formulas.Topics range from vital pre-calculus review to traditional calculus first-course content.Sample problems with solutions and a 50-problem chapter are ideal for self-testing.Fully explained examples with step-by-step solutions.
  right triangle trigonometry solving word problems answer key: CK-12 Trigonometry - Second Edition CK-12 Foundation, 2011-10-14 CK-12's Trigonometry-Second Edition is a clear presentation of trigonometry for the high school student. Its 6 chapters cover the following topics: Right Triangles and an Introduction to Trigonometry, Graphing Trigonometric Functions, Trigonometric Identities and Equations, Inverse Trigonometric Functions, Triangles and Vectors, and The Polar System.
  right triangle trigonometry solving word problems answer key: Schaum's Outline of Theory and Problems of Trigonometry Frank Ayres, Robert E. Moyer, 1999 This edition reflects the changes in the trigonometry curriculum that have taken place between 1993 and 1998. Following the rise of the scientific calculator, this revision updates the book by keeping calculator usage in place of outdated material on logarithms, discarding irrelevant material.
  right triangle trigonometry solving word problems answer key: 103 Trigonometry Problems Titu Andreescu, Zuming Feng, 2006-03-04 * Problem-solving tactics and practical test-taking techniques provide in-depth enrichment and preparation for various math competitions * Comprehensive introduction to trigonometric functions, their relations and functional properties, and their applications in the Euclidean plane and solid geometry * A cogent problem-solving resource for advanced high school students, undergraduates, and mathematics teachers engaged in competition training
  right triangle trigonometry solving word problems answer key: The Software Encyclopedia 2000 Bowker Editorial Staff, 2000-05
  right triangle trigonometry solving word problems answer key: Street-Fighting Mathematics Sanjoy Mahajan, 2010-03-05 An antidote to mathematical rigor mortis, teaching how to guess answers without needing a proof or an exact calculation. In problem solving, as in street fighting, rules are for fools: do whatever works—don't just stand there! Yet we often fear an unjustified leap even though it may land us on a correct result. Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often hands us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation. In Street-Fighting Mathematics, Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge—from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Illustrating each tool with numerous examples, he carefully separates the tool—the general principle—from the particular application so that the reader can most easily grasp the tool itself to use on problems of particular interest. Street-Fighting Mathematics grew out of a short course taught by the author at MIT for students ranging from first-year undergraduates to graduate students ready for careers in physics, mathematics, management, electrical engineering, computer science, and biology. They benefited from an approach that avoided rigor and taught them how to use mathematics to solve real problems. Street-Fighting Mathematics will appear in print and online under a Creative Commons Noncommercial Share Alike license.
  right triangle trigonometry solving word problems answer key: Handbook of Mathematical Functions Milton Abramowitz, Irene A. Stegun, 1965-01-01 An extensive summary of mathematical functions that occur in physical and engineering problems
  right triangle trigonometry solving word problems answer key: Problem-Solving Strategies Arthur Engel, 2008-01-19 A unique collection of competition problems from over twenty major national and international mathematical competitions for high school students. Written for trainers and participants of contests of all levels up to the highest level, this will appeal to high school teachers conducting a mathematics club who need a range of simple to complex problems and to those instructors wishing to pose a problem of the week, thus bringing a creative atmosphere into the classrooms. Equally, this is a must-have for individuals interested in solving difficult and challenging problems. Each chapter starts with typical examples illustrating the central concepts and is followed by a number of carefully selected problems and their solutions. Most of the solutions are complete, but some merely point to the road leading to the final solution. In addition to being a valuable resource of mathematical problems and solution strategies, this is the most complete training book on the market.
  right triangle trigonometry solving word problems answer key: Challenging Problems in Geometry Alfred S. Posamentier, Charles T. Salkind, 2012-04-30 Collection of nearly 200 unusual problems dealing with congruence and parallelism, the Pythagorean theorem, circles, area relationships, Ptolemy and the cyclic quadrilateral, collinearity and concurrency and more. Arranged in order of difficulty. Detailed solutions.
  right triangle trigonometry solving word problems answer key: Solving Mathematical Problems Terence Tao, 2006-07-28 Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the tactics involved in solving mathematical problems at the Mathematical Olympiad level. With numerous exercises and assuming only basic mathematics, this text is ideal for students of 14 years and above in pure mathematics.
  right triangle trigonometry solving word problems answer key: Acing the New SAT Math Thomas Hyun, 2016-05-01 SAT MATH TEST BOOK
  right triangle trigonometry solving word problems answer key: Applied Mathematics for the Petroleum and Other Industries Mildred Gerding, 1996-01-01 The original objective of this book was to supply a basic reference work to be used principally for review by field employees of the oil and gas industry. The petroleum industry had a definaite need for a comprehensive work that emphasized the application of math to field and shop work. IThis new edition is now geared towards using this book as a rext as well as a reference. - page v.
  right triangle trigonometry solving word problems answer key: The Shame Machine Cathy O'Neil, 2022-03-22 A TIMES BOOK OF THE YEAR Shame is being weaponized by governments and corporations to attack the most vulnerable. It's time to fight back Shame is a powerful and sometimes useful tool. When we publicly shame corrupt politicians, abusive celebrities, or predatory corporations, we reinforce values of fairness and justice. But as best-selling author Cathy O'Neil argues in this revelatory book, shaming has taken a new and dangerous turn. It is increasingly being weaponized -- used as a way to shift responsibility for social problems from institutions to individuals. Shaming children for not being able to afford school lunches or adults for not being able to find work lets us off the hook as a society. After all, why pay higher taxes to fund programmes for people who are fundamentally unworthy? O'Neil explores the machinery behind all this shame, showing how governments, corporations and the healthcare system capitalize on it. There are damning stories of rehab clinics, reentry programs, drug and diet companies, and social media platforms -- all of which profit from 'punching down' on the vulnerable. Woven throughout The Shame Machine is the story of O'Neil's own struggle with body image and her recent weight-loss surgery, which awakened her to the systematic shaming of fat people seeking medical care. With clarity and nuance, O'Neil dissects the relationship between shame and power. Whom does the system serve? How do current incentive structures perpetuate the shaming cycle? And, most important, how can we all fight back?
  right triangle trigonometry solving word problems answer key: 411 SAT Algebra and Geometry Questions , 2006 In order to align the SAT with the math curriculum taught in high schools, the SAT exam has been expanded to include Algebra II materials. 411 SAT Algebra and Geometry Questions is created to offer you a rigorous preparation for this vital section. If you are planning to take the SAT and need extra practice and a more in-depth review of the Math section, here's everything you need to get started. 411 SAT Algebra and Geometry Questions is an imperative study tool tailored to help you achieve your full test-taking potential. The most common math skills that you will encounter on the math portion of the SAT are covered in this book. Increase your algebra and geometry skills with proven techniques and test your grasp of these techniques as you complete 411 practice questions, including a pre- and posttest. Follow up by reviewing our comprehensive answer explanations, which will help measure your overall improvement. The questions are progressively more difficult as you work through each set. If you can handle the last question on each set, you are ready for the SAT! Book jacket.
  right triangle trigonometry solving word problems answer key: Teaching Mathematics in Grades 6 - 12 Randall E. Groth, 2012-08-10 Teaching Mathematics in Grades 6 - 12 by Randall E. Groth explores how research in mathematics education can inform teaching practice in grades 6-12. The author shows preservice mathematics teachers the value of being a researcher—constantly experimenting with methods for developing students' mathematical thinking—and connecting this research to practices that enhance students' understanding of the material. Ultimately, preservice teachers will gain a deeper understanding of the types of mathematical knowledge students bring to school, and how students' thinking may develop in response to different teaching strategies.
  right triangle trigonometry solving word problems answer key: College Algebra & Trigonometry Julie Miller, Donna Gerken, 2016-01-04 Julie Miller wrote her developmental math series because students were coming into her Precalculus course underprepared. They weren’t mathematically mature enough to understand the concepts of math nor were they fully engaged with the material. She began her developmental mathematics offerings with intermediate algebra to help bridge that gap. The Precalculus series is a carefully constructed end to that bridge that uses the highly effective pedagogical features from her fastest growing developmental math series. What sets Julie Miller’s series apart is that it addresses course issues through an author-created digital package that maintains a consistent voice and notation throughout the program. This consistency--in videos, PowerPoints, Lecture Notes, and Group Activities--coupled with the power of ALEKS and Connect Hosted by ALEKS, ensures that students master the skills necessary to be successful in Precalculus and can carry them through to the calculus sequence.
  right triangle trigonometry solving word problems answer key: The Stanford Mathematics Problem Book George Polya, Jeremy Kilpatrick, 2013-04-09 Based on Stanford University's well-known competitive exam, this excellent mathematics workbook offers students at both high school and college levels a complete set of problems, hints, and solutions. 1974 edition.
  right triangle trigonometry solving word problems answer key: Seberson Method: New SAT® Vocabulary Workbook Katya Seberson, 2020-02-25 Further your SAT vocabulary knowledge to get farther down the road to success This SAT vocabulary workbook helps students master more than 700 words that frequently appear in the SAT's reading, writing, and essay sections. The book's approach reflects changes made to the test in recent years, focusing on understanding vocabulary more than rote memorization. It's a modern workbook designed to give students the edge needed to improve their SAT scores. 145 short lessons—Each lesson features a theme to help contextualize vocabulary and concludes with a mini quiz to test understanding. Practical organization—Chapters focus on different elements of the SAT, including words for reading topics like history and science, transition words, and commonly confused words. Learning that lasts—With extra tips for retention, this focused approach works equally well for students who are taking the test in a week or in a year. Perfect for summer learning—This guide makes a great summer workbook for students planning to take the SAT this coming year who want to get a head start on studying before heading back to school. Get the ideal resource for students looking to master SAT vocabulary.
  right triangle trigonometry solving word problems answer key: Master Math Brita Immergut, 2009 Get ready to master the unknown number! Master Math: Solving Word Problems is a comprehensive reference guide that explains and clarifies the difficulties people often face with word problems, in a simple, easy-to-follow style and format. Beginning with the most basic types of word problems and progressing through to the more advanced, Solving Word Problems shows you how to focus first on the words in the problem, and then on the numbers, breaking down the problem into smaller segments to help you work through. Using familiar situations from everyday life such as percents and discounts, interest, motion and speed, and probability, each type of word problem is taught using step-by-step procedures, solutions, and examples. And end-of-chapter problems will help you practice what you learned. A complete table of contents and a comprehensive index enable you to quickly find specific topics, and the approachable style and format facilitate an understanding of what can be intimidating and tricky skills. Perfect for both students who need some extra help or rusty professionals who want to brush up, Solving Word Problems will help you master everything from simple equations and percents to statistics and probability!
  right triangle trigonometry solving word problems answer key: Helping Children Learn Mathematics National Research Council, Division of Behavioral and Social Sciences and Education, Center for Education, Mathematics Learning Study Committee, 2002-07-31 Results from national and international assessments indicate that school children in the United States are not learning mathematics well enough. Many students cannot correctly apply computational algorithms to solve problems. Their understanding and use of decimals and fractions are especially weak. Indeed, helping all children succeed in mathematics is an imperative national goal. However, for our youth to succeed, we need to change how we're teaching this discipline. Helping Children Learn Mathematics provides comprehensive and reliable information that will guide efforts to improve school mathematics from pre-kindergarten through eighth grade. The authors explain the five strands of mathematical proficiency and discuss the major changes that need to be made in mathematics instruction, instructional materials, assessments, teacher education, and the broader educational system and answers some of the frequently asked questions when it comes to mathematics instruction. The book concludes by providing recommended actions for parents and caregivers, teachers, administrators, and policy makers, stressing the importance that everyone work together to ensure a mathematically literate society.
  right triangle trigonometry solving word problems answer key: New General Mathematics for Junior Secondary Schools Murray Macrae, A. O. Kalejaiye, Z. I. Chima, G. U. Gaba, M. O. Ademosu, 2008-06-03 This well-established series, the most popular in Nigeria, has been fully revised to reflect recent developments in mathematics education at junior secondary level and the views of the many users of the books. It has expecially been revised to fully cover the requirements of the new NERDC Universal Basic Education Curriculum.
  right triangle trigonometry solving word problems answer key: Which One Doesn't Belong? Christopher Danielson, 2019-02-12 Talking math with your child is simple and even entertaining with this better approach to shapes! Written by a celebrated math educator, this innovative inquiry encourages critical thinking and sparks memorable mathematical conversations. Children and their parents answer the same question about each set of four shapes: Which one doesn't belong? There's no one right answer--the important thing is to have a reason why. Kids might describe the shapes as squished, smooshed, dented, or even goofy. But when they justify their thinking, they're talking math! Winner of the Mathical Book Prize for books that inspire children to see math all around them. This is one shape book that will both challenge readers' thinking and encourage them to think outside the box.--Kirkus Reviews, STARRED review
  right triangle trigonometry solving word problems answer key: Mathematics Framework for California Public Schools California. Curriculum Development and Supplemental Materials Commission, 1999
  right triangle trigonometry solving word problems answer key: The Pythagorean Theorem Eli Maor, 2019-11-19 Frontmatter --Contents --List of Color Plates --Preface --Prologue: Cambridge, England, 1993 --1. Mesopotamia, 1800 BCE --Sidebar 1: Did the Egyptians Know It? --2. Pythagoras --3. Euclid's Elements --Sidebar 2: The Pythagorean Theorem in Art, Poetry, and Prose --4. Archimedes --5. Translators and Commentators, 500-1500 CE --6. François Viète Makes History --7. From the Infinite to the Infinitesimal --Sidebar 3: A Remarkable Formula by Euler --8. 371 Proofs, and Then Some --Sidebar 4: The Folding Bag --Sidebar 5: Einstein Meets Pythagoras --Sidebar 6: A Most Unusual Proof --9. A Theme and Variations --Sidebar 7: A Pythagorean Curiosity --Sidebar 8: A Case of Overuse --10. Strange Coordinates --11. Notation, Notation, Notation --12. From Flat Space to Curved Spacetime --Sidebar 9: A Case of Misuse --13. Prelude to Relativity --14. From Bern to Berlin, 1905-1915 --Sidebar 10: Four Pythagorean Brainteasers --15. But Is It Universal? --16. Afterthoughts --Epilogue: Samos, 2005 --Appendixes --Chronology --Bibliography --Illustrations Credits --Index.
  right triangle trigonometry solving word problems answer key: Practical Math Success in 20 Minutes a Day , 2012 This book guides you through pre-algebra, algebra, and geometry -- the fundamental concepts you're striving to conquer, or the ones you once learned but may have forgotten -- in just 20 easy steps.
  right triangle trigonometry solving word problems answer key: Trigonometric Delights E. Maor, 1998 In this book, Maor rejects the usual arid descriptions of the sine and cosine functions and their trigonometric relatives. He brings the subject to life in a compelling blend of mathematics, history , and biography. Form the 'proto-trigonometry' of the Egyptian pyramid builders to Renaissance Europe's quest for more accurate artillery, from the earliest known trigonometric table......
  right triangle trigonometry solving word problems answer key: Plane Trigonometry Sidney Luxton Loney, 1893
Applications of Right Triangles and Trig Functions - Math Plane
Trigonometry Word Problems Applications of Right Triangles and Trig Functions Includes angle of elevation and depression, examples, step-by-step solutions, and more…

Name ÿÿ Period Section 4.3 Worksheet - Right Triangle …


Right Triangle Trigonometry Solving Word Problems
Right Triangle Trigonometry Solving Word Problems Worksheet Answer Key: Algebra and Trigonometry Jay P. Abramson,Valeree Falduto,Rachael Gross (Mathematics teacher),David …

Right Triangle Trig Missing Sides and Angles - Kuta Software
Right Triangle Trig. - Finding Missing Sides and AnglesDate_____ Period____ Find the measure of each angle indicated. Round to the nearest tenth. 1) 13 12 B A C θ 2) 4 13 A B C θ 3) 9 6 A …

Mr. Yang's Teacher Website - Calculus AB


Dr. Neal, WKU MATH 117 Right Triangle Word Problems
Here we shall consider various word problems that provide some physical applications of the right-triangle trig formulas. and y = zsin. We often measure an angle from the ground upward …

Right Triangle Trigonometry Test Review - Math 2


Right Triangle Trigonometry Word Problems - Dave Goldsmith
Right Triangle Trigonometry Word Problems. 1. From the top of a lighthouse 160 feet above sea level, the angle of depression to a boat at sea is 25 degrees. To the nearest foot, what is the …

4.1 Right Triangle Trigonometry Word Problems Notes - Kara …


Unit 8 - Right Triangles & Trigonometry
Find the perimeter of the square: 24. The infield of a baseball field is a square with sides measuring 90 feet. A ball is thrown from third to first base is caught in I .2 seconds. Find the …

9-Solving Right Triangles - Kuta Software
Solving Right Triangles Date_____ Period____ Find the missing side. Round to the nearest tenth. 1) 6 x 72° 19.4 2) x 6 ...

RIGHT TRIANGLE TRIGONOMETRY - University of Houston
RIGHT TRIANGLE TRIGONOMETRY. The word Trigonometry can be broken into the parts Tri, gon, and metry, which means “Three angle measurement,” or equivalently “Triangle …

SOLVING RIGHT TRIANGLES WORKSHEET - Mr.


The Primary Trigonometric Ratios – Word Problems - MR.


Supplementary Trigonometry Exercise Problems - City …
Trig Section 1.3: Applying Right Triangles SHORT ANSWER. Solve the problem. 1) A 29 foot water slide has a 17 foot vertical ladder. How far is it along the ground from the end of the slide …

Section 13.1 - Right Triangle Trigonometry - Solving Right …
Answers to Worksheet #2 - Solving Right Triangles. 1) m A 53.1°, m B 36.9°, c 15 2) m B 52°, b 16, c 20.3. 3) m A 48.2°, m B 41.8°, a 2.2 4) m B 31°, b 8.2, a 13.7. 5) m A 63.4°, m B 26.6°, c …

PreCalc 5.1 Right Triangle Trig Review Name ...
V. Solve each word problem below. Draw a triangle and label the side lengths and angle measures before you decide which trig function to use. Some problems may require you to …

Right Triangle Trigonometry Word Problem Examples
Right Triangle Trigonometry Word Problem Examples. Example 1. If B = 42° and a = 12, find c. From the triangle at the right, you know the measure of an angle and the side adjacent. You …

Right Triangle Trigonometry Solving Word Problems …
Right Triangle Trigonometry Solving Word Problems Worksheet Answer Key: Algebra and Trigonometry Jay P. Abramson,Valeree Falduto,Rachael Gross (Mathematics teacher),David …

PROBLEMS WORKSHEET PYTHAGOREAN THEOREM WORD
PYTHAGOREAN THEOREM WORD. PROBLEMS WORKSHEET. 1. A swimmer starts at a point on the shore and swims 200 meters due east. Then, the swimmer. turns and swims 300 meters due north. What is the shortest distance for the swimmer to return. to the starting point to the nearest metre? 2. On the rooftop of a skyscraper, a square garden is designed with ...

Right Triangle Trigonometry
%PDF-1.6 %âãÏÓ 183 0 obj > endobj 235 0 obj >/Filter/FlateDecode/ID[0E9FEC4EC85E4412B04AE8B87E8F720B>]/Index[183 115]/Info 182 0 R/Length 191/Prev 544224/Root 184 ...

Trigonometry - Hard Problems Q&A - MathGuy.US
Trigonometry – Hard Problems Based on the illustration at right, we get the following: tan à L 90 200.45 à Ltan ? 5.45 L° The angle 7 9 is in Q2, but tangent is defined only in Q1 and Q4. Further, tan 7 9 0 in Q2.

Mathematics (Linear) 1MA0 TRIGONOMETRY - Maths Genie
TRIGONOMETRY Materials required for examination Items included with question papers ... ABC is a right-angled triangle. Angle B = 90 . Angle A = 36 . AB = 8.7 cm. ... Calculate the size of angle a in this right-angled triangle. Give your answer correct to 3 significant figures. ..... (3 marks) xq P 12 cm R Q 4.5 cm a 6 m 5 m Diagram ...

Similar Triangles and Ratios - Math Plane
Solving "similar triangle/ratio" problems: 1) Draw picture 2) Split triangles 3) Solve proportion 4) Check answer Example: Solution: Step 1: Step 3: Solve the proportion x Step 4: Check answer 9.64 = 34 +5 91 x=34 3.64 1.56 9.64 5.66 1.56 3.64 49.5 8.25 8.25 x = 3.64 feet A 6-foot tall man casts a shadow 4 feet. And, a house casts a shadow 33 feet.

Unit 3 Ans - Houston Independent School District
Solving Right Triangles: If we know two pieces of information of a right triangle, we can solve that triangle – that is find all the missing information from that triangle. For this section, we will make some generalizations: that the triangle will be labeled ! "ABC where the side opposite angle A is labeled a, the side

Right triangle trigonometry word problems worksheet …
Right triangle trigonometry word problems worksheet answers question 1 :The elevation angle of the upper part of the building at a distance of 50 m from its foot on a horizontal plane is 60 degrees. find the height of the building. question 2 : a scale placed against a wall to reach the upper part of the wall height 6 m and the staircase is inclined to a angle of 60

Right Triangle Trig Evaluating Ratios - Kuta Software
Right Triangle Trig. - Evaluating Trig. Ratios Date_____ Period____ Find the value of the trig function indicated. 1) sec θ 17 8 15 θ 2) sec θ 13 5 12 θ 3) cot θ 5 3 4 θ 4) csc θ 17 15 8 θ 5) csc θ 16 2 24 8 θ 6) cos θ 21 7 14 2 θ 7) cot θ 25 15 20 θ 8) tan θ 24 22 2 23 θ 9) tan θ 10 6 8 θ 10) cot θ 6 4 2 5 θ 11) tan θ 5 4 ...

Right Triangle Trig Missing Sides and Angles - Kuta Software
Right Triangle Trig. - Finding Missing Sides and AnglesDate_____ Period____ Find the measure of each angle indicated. Round to the nearest tenth. 1) 13 12 B A C θ 2) 4 13 A B C θ 3) 9 6 A B C θ 4) 11.9 10 B A C θ 5) 7.7 14 A B C θ 6) 5 B 4 A C θ 7) 11 4.4 A B C θ 8) 3 3 B C A θ Find the measure of each side indicated. Round to the ...

4.8 Solving Problems with Trigonometry
More Right Triangle Problems We close this first of two trigonometry chapters by revisiting some of the applica-tions of Section 4.2 (right triangle trigonometry) and Section 4.4 (sinusoids). An angle of elevationis the angle through which the eye moves up from horizontal to look at something above, and an angle of depressionis the angle ...

5.8 Solving 3-D Trig Problems.notebook - Mr. Hardy's Virtual …
5.8 Solving 3­D Trig Problems.notebook 1 April 16, 2015 5.8 Solving 3‐D Trig Problems 3-dimensional trigonometry problems often involve constructing 2 or 3 triangles and solving for multiple sides and angles in order to reach the final answer. Typically drawing the triangle correctly is the most difficult aspect of these questions.

Right Triangle Trigonometry Solving Word Problems …
Right Triangle Trigonometry Solving Word Problems Answer Key Right Triangle Trigonometry: Solving Word Problems – Answer Key & ... Mastering right triangle trigonometry word problems is a crucial skill in mathematics and its applications. By following the systematic approach outlined above, practicing diligently, and working through diverse ...

Chapter 1 – Introduction to Trigonometry Answer Key
Chapter 1 – Introduction to Trigonometry Answer Key CK-12 Trigonometry Concepts 2 1.2 Identifying Sets of Pythagorean Triples Answers 1. Yes 2. No 3. No 4. Yes 5. No 6. Yes 7. No 8. No 9. Yes 10. Yes 11. Yes 12. No 13. It will allow you to avoid using the Pythagorean Theorem when trying to find missing sides of right triangles in problems. 14.

Right Triangle Trigonometry Solving Word Problems …
Right Triangle Trigonometry: Solving Word Problems – A Comprehensive Guide Introduction: Understanding the Fundamentals Right triangle trigonometry is a powerful tool used to solve problems involving triangles with one 90-degree angle. The foundation lies in understanding the three primary trigonometric functions: sine, cosine, and tangent.

9-Inverse Trigonometric Ratios - Kuta Software
26) Draw and label all three sides of a right triangle that has a 40 ° angle and a hypotenuse of 10 cm. Sides are: 10 cm, 6.4 cm, and 7.7 cm.-2-Create your own worksheets like this one with Infinite Geometry. Free trial available at KutaSoftware.com

Algebra 2/Trig Name: Unit 8 Notes Packet Date: Period
ANALYTIC TRIGONOMETRY is an extension of right triangle trigonometry. It takes place on the x-y plane. For, trigonometry as it is actually used in calculus and physics, is not about solving triangles. It becomes the mathematical description of things that rotate or vibrate, such as light, sound, the paths of planets

Right Triangle Trigonometry - Finding Side Lengths
Right Triangle Trigonometry - Finding Side Lengths Name_____ Date_____-1-Step One: On each diagram below, label each side according to the position of the reference angle. Your options are: Opposite, Adjacent, Hypotenuse. 1) 14 x 52° 2) 12 x 16° 3) x 19 22° 4) 13x 69°

Right Triangle Trigonometry Solving Word Problems …
Worksheet Answer Key Right Triangle Trigonometry: Solving Word Problems Worksheet Answer Key Are you struggling with right triangle trigonometry word problems? Do you feel lost navigating the angles, sides, and applications of sine, cosine, and tangent? You've come to the right place! This comprehensive guide provides not just a simple answer ...

Right Triangle Trigonometry: Solving Word Problems
Trigonometry is used on a daily basis in the workplace. Since trigonometry means "triangle measure", any profession that deals with measurement deals with trigonometry as well. Carpenters, construction workers and engineers, for example, must possess a thorough understanding of trigonometry. Right Triangle Trigonometry: Solving Word Problems

9 Right Triangles and Trigonometry - Big Ideas Learning
Standard Position for a Right Triangle In unit circle trigonometry, a right triangle is in standard position when: 1. The hypotenuse is a radius of the circle of radius 1 with center at the origin. 2. One leg of the right triangle lies on the x-axis. 3. The other leg of the right triangle is perpendicular to the x-axis. CCore ore CConceptoncept ...

Special Right Triangles - Big Ideas Learning
9.2 Special Right Triangles 461 Work with a partner. a. One type of special right triangle is a 45°- 45 °- 90 ° triangle. i. Construct a right triangle with acute angle measures of 45°. ii. Find the exact ratios of the side lengths. AB — BC = AB — AC = AC — BC = iii. Repeat parts (i) and (ii) for se veral other 45°- 45 °- 90 ...

Trigonometry - Angles of Elevation & Depression …
Q26 . The base angles of an isosceles triangle are 41º. If the equal sides are 9.5 cm long, find the length of the base. Q27 . The base of an isosceles triangle is 18cm long and the perpendicular height is ... Trigonometry - Angles of Elevation & Depression Worksheet Author: info@xceleratemath.com Created Date: 5/31/2021 5:18:09 PM ...

Right Triangle Trigonometry Word Problem Examples
Right Triangle Trigonometry Word Problem Examples Example 1 If B = 42° and a = 12, find c. From the triangle at the right, you know the measure of an angle and the side adjacent. You want to know the measure of the hypotenuse. The cosine function relates the side adjacent to the angle and the hypotenuse. cos B = a c cos = side adjacent hypotenuse

I. Model Problems. II. Identifying Opposite, Adjacent and …
Part I Model Problems sin( ) opposite B hypotenuse cos( ) adjacent B hypotenuse tan( ) opposite B adjacent Model Problem 1 ) Identify The side adjacent, opposite to angle x and the hypotenuse Adjacent to x : A Opposite X : B Hypotenuse : C Model Problem 2) What is sin( k ) , cos( k ) and tan(k )? Use SOHCAHTOA

Right Triangle Trigonometry Solving Word Problems …
Right Triangle Trigonometry: Solving Word Problems – A Comprehensive Guide Introduction: Understanding the Fundamentals Right triangle trigonometry is a powerful tool used to solve problems involving triangles with one 90-degree angle. The foundation lies in understanding the three primary trigonometric functions: sine, cosine, and tangent.

Geometry: Similar Triangles—Explanation & Practice - Weebly
• Each has a right angle (90°). These angles are indicated by squares. • The angles labeled 1 and 2 are equal because they are vertical angles. • The angles labeled 3 and 4 are also equal. If two triangles have two equal angles, the third angles are equal because the sum of the angles 4 yd. in each triangle must total 180°.) Answer Key

6.1 Basic Right Triangle Trigonometry - Rochester Institute …
3 7. Right triangle: a triangle with a right angle (an angle of 𝜋 2 radians) 8. Isosceles triangle: a triangle with exactly two sides of equal length 9. Equilateral triangle: a triangle with all three sides of equal length 10. Hypotenuse: side opposite the right angle, side c in the diagram above 11. 2Pythagorean Theorem: = 2+ Example 1: A right triangle has a hypotenuse length of 5 inches.

Right Triangle Trigonometry - Big Ideas Learning
Section 9.1 Right Triangle Trigonometry 463 Evaluating Trigonometric Functions In a right triangle, θ is an acute angle and sin θ = 4— 7. Evaluate the other fi ve trigonometric functions of θ. SOLUTION Step 1 Draw a right triangle with acute angle θ such that the leg opposite θ has length 4 and the hypotenuse has length 7.

Unit 4.2 Right Triangles/ Vectors
The trigonometric functions of a right triangle, with an angle θ, are defined by ratios of two sides of the triangle. The sides of the right triangle are: OPP the side opposite the angle θ ADJ the side adjacent to the angle θ HYP is the hypotenuse of the right triangle.

Right Triangle Trigonometry Solving Word Problems …
Right Triangle Trigonometry: Solving Word Problems – A Comprehensive Guide Introduction: Understanding the Fundamentals Right triangle trigonometry is a powerful tool used to solve problems involving triangles with one 90-degree angle. The foundation lies in understanding the three primary trigonometric functions: sine, cosine, and tangent.

Right Triangle Trigonometry Solving Word Problems …
This comprehensive guide breaks down word problems into manageable steps, making trigonometry accessible and engaging. Learn through real-world examples and practice problems. Keywords: right triangle trigonometry, word problems, trigonometry, math, geometry, sine, cosine, tangent, problem-solving,

Right Triangle Applications - Radford University
G.7 The student will solve practical problems involving right triangles by using the Pythagorean Theorem, properties of special right triangles, and right triangle trigonometry. Solutions will be expressed in radical form or as decimal approximations. NCTM Standards Use trigonometric relationships to determine lengths and angle measures.

The Journey to Discover Right Triangles and Trigonometry in …
Jun 9, 2012 · 1. The parts of a right triangle include two legs, and a hypotenuse. The hypotenuse is always the longest side of the right triangle because it is the side across from the largest angle in the right triangle, the 90 degree angle. Since one angle of a right triangle will always be 90 degrees, we can make a conjecture that the acute angles

Holt Mcdougal Solving Right Triangles Answer Key (PDF)
Holt Mcdougal Solving Right Triangles Answer Key: ... Trigonometry In Precalculus we approach trigonometry by first introducing angles and the unit circle as opposed to the right triangle approach more commonly used in College Algebra and Trigonometry courses Chapter 5 Trigonometric Functions

The Primary Trigonometric Ratios – Word Problems - Tripod
B. Solving problems involving two right triangles in two dimensions. To solve a problem involving two right triangles using trigonometry, • draw and label a diagram showing the given information, and the length or angle measure to be found • identify the two triangles that can be used to solve the problem, and plan how to use each triangle

Right Triangle Trigonometry: Solving Word Problems
Trigonometry is used on a daily basis in the workplace. Since trigonometry means "triangle measure", any profession that deals with measurement deals with trigonometry as well. Carpenters, construction workers and engineers, for example, must possess a thorough understanding of trigonometry. Right Triangle Trigonometry: Solving Word Problems

Performance Based Learning and Assessment Task How Tall Is …
• The use of sine, cosine, tangent in solving right triangle trigonometry word problems. 3 . Introduction: Setting Up the Mathematical Task ... a quick answer. As our summer intern, you need to provide the answer. As is typical, your ... the real-world application of right triangle trigonometry to solve problems and obtain hard-

9 Right Triangles and Trigonometry - Big Ideas Learning
Standard Position for a Right Triangle In unit circle trigonometry, a right triangle is in standard position when: 1. The hypotenuse is a radius of the circle of radius 1 with center at the origin. 2. One leg of the right triangle lies on the x-axis. 3. The other leg of the right triangle is perpendicular to the x-axis. Core Concept x y 0.5 − ...

NAME DATE PERIOD 8-4 Word Problem Practice
were both solving the same trigonometry problem. However, after they finished their computations, Melinda said the answer was 52 sin 27° and Walter said the answer was 52 cos 63°. Could they both be correct? Explain. 4. LINES Jasmine draws line m on a coordinate plane. m x y-5 O 5 5 What angle does m make with the x-axis? Round your answer to ...

8.1 Pythagorean Theorem and Pythagorean Triples
Chapter 8 – Right Triangle Trigonometry Answer Key CK-12 Geometry Concepts 3 17. right 18. obtuse 19. obtuse 20. acute 21. obtuse 22. One way is to use the distance formula to find the distances of all three sides and then use the converse of the Pythagorean Theorem. The second way would be to find the

Quarter 4-Module 9 Solve Problems Involving Oblique …
Now that we can solve oblique triangles, we can use this concept in solving real-life problems. To solve word problems involving oblique triangle, the following steps are suggested: 1. Draw a picture representing the situation in the problem. 2. Label the picture with the give information and a variable for the unknown measure/s of the part/s. 3.

WORKSHEET ­ General 2 Mathematics - JSCHS MATHEMATICS …
MM5 – Apps of Trigonometry ­ Harder applications (2 triangle questions) ­ Bearings/Sine Cosine Rules Teacher: PETER HARGRAVES Source: HSC exam questions Exam Equivalent Time: 87 minutes Worked Solutions: Included Note: Each question has designated marks.

Unit 2 Grade 10 Applied Trigonometry - OAME
5 Solving Right-Angled Triangles (Part 1) • Determine the measures of the sides and angles in right-angled triangles, using the primary trigonometric ratios and the Pythagorean theorem. MT2.02, MT2.03 CGE 3f 6 Solving Right-Angled Triangles (Part 2) • Students make and use clinometers to measure angles.

Right Triangle Trigonometry Solving Word Problems …
Right Triangle Trigonometry Solving Word Problems Answer Key: Pre-Calculus Workbook For Dummies Yang Kuang,Michelle Rose Gilman,Elleyne Kase,2011-04-12 This hands on workbook helps students master basic pre calculus concepts and practice the types of problems they ll encounter in the

Right Triangle Trigonometry Solving Word Problems …
Worksheet Answer Key Right Triangle Trigonometry: Solving Word Problems Worksheet Answer Key Are you struggling with right triangle trigonometry word problems? Do you feel lost navigating the angles, sides, and applications of sine, cosine, and tangent? You've come to the right place! This comprehensive guide provides not just a simple answer ...

Right Triangle What You’ve Learned Trigonometry
Right Triangle Trigonometry Answers are in the back of the textbook. For intervention, direct students to: Solving Proportions Lesson 7-1: Example 3 Extra Skills, Word Problems, Proof Practice, Ch. 7 Proving Triangles Similar Lesson 7-3: Examples 1 and 2 Extra Skills, Word Problems, Proof Practice, Ch. 7 Similarity in Right Triangles Lesson 7-4 ...

Acellus Trigonometry INTERNATIONAL ACADEMY OF SCIENCE
3 Right Triangles: 45-45-90 31 Graphs of Sine, Cosine, and Tangent 4 Rationalizing the Denominator 32 Graph of Cosine 5 Degrees, Minutes, Seconds 33 Graph of Tangent Unit 2 - Right Triangle Trigonometry 34 Graph of Secant and Cosecant 6 Sine, Cosine, and Tangent 35 Graph of Cotangent 7 Word Problems with Sine, Cosine, and Tangent 36 Amplitude