...th Trying trig Everything you need to Know By: Noah Gregory subject Page Radians & Degree Measure 3 Unite Circle 4 Right Triangle Trig 5-7 trig functions of any angle 8-10 graphs 11-15 using fundamental trig identities 16-17 verifying trig identities 18-20 solving trig equations 21-23 sum & difference formulas 24 law of sines 25-27 laws of cosines 28-29 vectors 30-31 Definitions 32-33 Radians & Degree Measure Converting radians to degrees: To convert radians to degrees, we make use of the fact that p radians equals one half circle, or 180º. [pic] This means that if we divide radians by p, the answer is the number of half circles. Multiplying this by 180º will tell us the answer in degrees. So, to convert radians to degrees, multiply by 180/p, like this: [pic] To convert degrees to radians, first find the number of half circles in the answer by dividing by 180º. But each half circle equals p radians, so multiply the number of half circles by p. Example 1 (p= Pie) 10º in radians would be 18 Radians. First put your degree over 1 R= 10°/1 (p/180°) Next multiply & divide & you will get 18p ------------------------------ Example 2 1.4 Radians would be 80.2° put your radian over 1 D= 1.4/1 (180°/p) Next multiply & divide & you will get 80.2 ° Unite Circle ...
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...Project in Math Chapter 6 Circular Functions and Trigonometry Submitted to: Mrs. Velasco Submitted by: Angelo Jet Manarin IV – Bravo Angles & Their Measures A definition of an angle would be that an angle is the union of two rays that have the same endpoint. The sides of the angles are the two rays, while the vertex is their common endpoint. stands for an angle. You can put it in front of three letters which represent points. The first and third letters represent points on each of the rays that form one of the sides. The middle letter represents the vertex. As you can see in the diagram, each point is represented in the written form. The letters can go either way - that is, first and last letter are interchangeable. So, that angle could either be ABC or CBA. Since B is the vertex, it is always in the middle of the two letters. You can also name an angle by just the letter of its vertex. So, for the example in the picture, the angle could also be labeled B. That's only if there are no other angles that share the same vertex. There is a third way to label angles. In the third way, each angle is designated with a number, so the example could be labeled 1 or 2 or whatever you wanted. Angles are measured in degrees. The number of degrees tell you how wide open the angle is. You can measure angles with a protracter, and you can buy them at just about any store that carries school items. Degrees are marked by a ° symbol. For those of you whose browsers can't interpret that, a...
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...Trigonometry Review with the Unit Circle: All the trig. you’ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs: domain, range and transformations. Angle Measure Angles can be measured in 2 ways, in degrees or in radians. The following picture shows the relationship between the two measurements for the most frequently used angles. Notice, degrees will always have the degree symbol above their measure, as in “452 ° ”, whereas radians are real number without any dimensions, so the number “5” without any symbol represents an angle of 5 radians. An angle is made up of an initial side (positioned on the positive x-axis) and a terminal side (where the angle lands). It is useful to note the quadrant where the terminal side falls. Rotation direction Positive angles start on the positive x-axis and rotate counterclockwise. Negative angles start on the positive x-axis, also, and rotate clockwise. Conversion between radians and degrees when radians are given in terms of “ π ” DEGREES RADIANS: The official formula is θ ⋅ π 180 = θ radians Ex. Convert 120 into radians SOLUTION: 120 ⋅ π 180 = 2π radians 3 RADIANS DEGREES: The conversion formula is θ radians ⋅ 180 π =θ Ex. Convert π 5 into degrees. SOLUTION: π 180 180 ⋅ = = 36 5 π 5 For your own reference, 1 radian ≈ 57.30 A radian is defined by the radius of a circle. If you measure off...
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...UNIT CIRCLE TRIGONOMETRY The Unit Circle is the circle centered at the origin with radius 1 unit (hence, the “unit” circle). The equation of this circle is x 2 + y 2 = 1 . A diagram of the unit circle is shown below: y 1 x2 + y2 = 1 x -2 -1 -1 1 2 -2 We have previously applied trigonometry to triangles that were drawn with no reference to any coordinate system. Because the radius of the unit circle is 1, we will see that it provides a convenient framework within which we can apply trigonometry to the coordinate plane. Drawing Angles in Standard Position We will first learn how angles are drawn within the coordinate plane. An angle is said to be in standard position if the vertex of the angle is at (0, 0) and the initial side of the angle lies along the positive x-axis. If the angle measure is positive, then the angle has been created by a counterclockwise rotation from the initial to the terminal side. If the angle measure is negative, then the angle has been created by a clockwise rotation from the initial to the terminal side. θ in standard position, where y Terminal side θ is positive: θ in standard position, where y θ is negative: θ Initial side Initial side x θ x Terminal side Unit Circle Trigonometry Drawing Angles in Standard Position Examples The following angles are drawn in standard position: 1. θ = 40 y 2. θ = 160 θ y x θ x y 3. θ = −320 θ x Notice that...
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...learn (1) how to find the trigonometric functions using right triangles, (2) compute the values of these functions for some special angles, and (3) solve model problems involving the trigonometric functions. First, let’s review some of the features of right triangles. A triangle in which one angle is 90◦ is called a right triangle. The side opposite to the right angle is called the hypotenuse and the remaining sides are called the legs of the triangle. Suppose that we are given an acute angle θ as shown in Figure 11.1. Note that a = 0 and b = 0. Figure 11.1 Associated with θ are three lengths, the hypotenuse , the opposite side, and the adjacent side. We define the values of the trigonometric functions of θ as ratios of the sides of a right triangle: sin θ = csc θ = opposite hypotenuse hypotenuse opposite = = b r r b cos θ = sec θ = adjacent hypotenuse hypotenuse adjacent = = a r r a tan θ = cot θ = opposite adjacent adjacent opposite = = b a a b √ where r = a2 + b2 (Pythagorean formula). The symbols sin, cos, sec, csc, tan, and cot are abbreviations of sine, cosine, secant, cosecant, tangent, and cotangent. The above ratios are the same regardless of the size of the triangle. That is, the trigonometric functions defined above depend only on the angle θ. To see this, consider 1 Figure 11.2. Figure 11.2 The triangles ∆ABC and ∆AB C are similar. Thus, For example, using the cosine function we have cos θ = |AB| |AB | = |AC| |AC | = |BC| |B C | . |AB| |AB | = . |AC|...
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...How JPEG works Introduction - Images An image is a 2D matrix of values Color images consist of 3 RGB matrices, Grayscale images consist of one This basic format is BMP Introduction - JPEG JPEG – Joint Photographic Expert Group Late 1980’s, early 1990’s Consumer level computers Relatively high processing power Low bandwidth. Goal: File size↓ Picture quality~ Voted as international standard in 1992 Photos Scanned text or images with sharp transitions Introduction - Motivation 2 Observations: A large majority of useful image content changes slowly across images Experiments show that the human eye is less sensitive to loss of high spatial frequencies Now let’s see how we can use these observations!!! Outline Phase 1: Color transform Phase 2: Divide the picture Phase 3: Conversion to frequency domain Phase 4: Quantization Phase 5: Entropy Coding Comparison to other formats Outline – phase 1 (lossless) Phase 1: Color transform Phase 2: Divide the picture Phase 3: Conversion to frequency domain Phase 4: Quantization Phase 5: Entropy Coding Comparison to other formats Color transform RGB->YCbCr Red RGB Image Green Blue Color transform RGB->YCbCr Transformation in the color space RGB has all the information, however, if we want to encode, we can’t decide which of the Red/Green/Blue is more important. Y – brightness (luma), Cb – Chromatic blue, Cr – Chromatic red. Only Y will show black/white TV. More bits are used for Y, less for the Cb/Cr...
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...Sketch a graph of the sound wave. Sketch the angle in standard position. 4. 55º 5. –150º 6. Find the measure of an angle between 0º and 360º coterminal with an angle of –110º in standard position. 7. Find the exact value of cos 300º and sin 300º. Write the measure in radians. Express the answer in terms of π. 8. 320º 9. 45º Write the measure in degrees. 10. [pic] radians 11. –[pic] radians 12. Find the degree measure of an angle of 4.23 radians. 13. Find the exact values of [pic] and [pic]. 14. Use the circle below. Find the length s to the nearest tenth. [pic] 15. A Ferris wheel has a radius of 80 feet. Two particular cars are located such that the central angle between them is 165º. To the nearest tenth, what is the length of the intercepted arc between those two cars on the Ferris wheel? Use the graph of y = sin θ to find the value of sin θ for each value of θ. 16. 270º [pic] 17. [pic][pic] radians [pic] 18. Find the period of the graph shown below. [pic] 19. Sketch one cycle of y = 4 sin 4θ. 20. Write the equation for the sine function shown below. [pic] Write a cosine function for the graph. 21. [pic] 22. Use the graph of [pic] to find the value of y = tan [pic]π. Round to the nearest tenth if necessary. If the tangent is undefined at that point, write undefined. [pic] 23. Sketch the graph of the tangent curve y = tan [pic][pic] in the interval from 0 to 2π. Graph...
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...7-3: COMPUTING THE VALUES OF TRIG FUNCTIONS In a right triangle, if one of the acute angles = 45(, then so does the other; and the triangle is isosceles and could have legs = 1, hypotenuse = [pic]. In a right triangle, if one of the acute angles is 30(, then the other is 60(; such a triangle could have a hypotenuse of 2 and legs of 1 and [pic]. Find the exact value of the six trigonometric functions of 45(, 30(, and 60(: | |Sine |Cosine |Tangent |Cotangent |Cosecant |Secant | |45(=(/4 | | | | | | | |30(=(/6 | | | | | | | |60(=(/3 | | | | | | | Find the exact value of each expression if ( = 60(; do not use a calculator: 1. [pic] 2. 3 csc ( 3. [pic] Find the exact value of each expression; do not use a calculator: 4. 4 sin 45( + 2 cos 30( 5. 5 tan 30( . sin 60( 6. 1 + sec2 45( - cos2 60( Use a calculator to find the approximate value of each expression; round to 2 decimal places: 7. cos 42( 8. sec 38( 9. csc 72( 10. [pic] (use radian...
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...College Trigonometry Version π Corrected Edition by Carl Stitz, Ph.D. Lakeland Community College Jeff Zeager, Ph.D. Lorain County Community College July 4, 2013 ii Acknowledgements While the cover of this textbook lists only two names, the book as it stands today would simply not exist if not for the tireless work and dedication of several people. First and foremost, we wish to thank our families for their patience and support during the creative process. We would also like to thank our students - the sole inspiration for the work. Among our colleagues, we wish to thank Rich Basich, Bill Previts, and Irina Lomonosov, who not only were early adopters of the textbook, but also contributed materials to the project. Special thanks go to Katie Cimperman, Terry Dykstra, Frank LeMay, and Rich Hagen who provided valuable feedback from the classroom. Thanks also to David Stumpf, Ivana Gorgievska, Jorge Gerszonowicz, Kathryn Arocho, Heather Bubnick, and Florin Muscutariu for their unwaivering support (and sometimes defense) of the book. From outside the classroom, we wish to thank Don Anthan and Ken White, who designed the electric circuit applications used in the text, as well as Drs. Wendy Marley and Marcia Ballinger for the Lorain CCC enrollment data used in the text. The authors are also indebted to the good folks at our schools’ bookstores, Gwen Sevtis (Lakeland CC) and Chris Callahan (Lorain CCC), for working with us to get printed copies to the students...
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...TLFeBOOK WHAT READERS ARE SAYIN6 "I wish I had had this book when I needed it most, which was during my pre-med classes. I t could have also been a great tool for me in a few medical school courses." Or. Kellie Aosley8 Recent Hedical school &a&ate "CALCULUS FOR THE UTTERLY CONFUSED has proven to be a wonderful review enabling me t o move forward in application of calculus and advanced topics in mathematics. I found it easy t o use and great as a reference for those darker aspects of calculus. I' Aaron Ladeville, Ekyiheeriky Student 'I1am so thankful for CALCULUS FOR THE UTTERLY CONFUSED! I started out Clueless but ended with an All' Erika Dickstein8 0usihess school Student "As a non-traditional student one thing I have learned is the Especially in importance of material supplementary t o texts. calculus it helps to have a second source, especially one as lucid and fun t o read as CALCULUS FOR THE UTTERtY CONFUSED. Anyone, whether you are a math weenie or not, will get something out of this book. With this book, your chances of survival in the calculus jungle are greatly increased.'I Brad &3~ker, Physics Student Other books i the Utterly Conhrsed Series include: n Financial Planning for the Utterly Confrcsed, Fifth Edition Job Hunting for the Utterly Confrcred Physics for the Utterly Confrred CALCULUS FOR THE UTTERLY CONFUSED Robert M. Oman Daniel M. Oman McGraw-Hill New York San Francisco Washington, D.C. Auckland Bogoth Caracas Lisbon...
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...iOREGON DEPARTMENT OF TRANSPORTATION GEOMETRONICS 200 Hawthorne Ave., B250 Salem, OR 97310 (503) 986-3103 Ron Singh, PLS Chief of Surveys (503) 986-3033 BASIC SURVEYING - THEORY AND PRACTICE David Artman, PLS Geometronics (503) 986-3017 Ninth Annual Seminar Presented by the Oregon Department of Transportation Geometronics Unit February 15th - 17th, 2000 Bend, Oregon David W. Taylor, PLS Geometronics (503) 986-3034 Dave Brinton, PLS, WRE Survey Operations (503) 986-3035 Table of Contents Types of Surveys ........................................................................................... 1-1 Review of Basic Trigonometry ................................................................... 2-1 Distance Measuring Chaining ................................................................... 3-1 Distance Measuring Electronic Distance Meters ................................... 4-1 Angle Measuring .......................................................................................... 5-1 Bearing and Azimuths ................................................................................ 6-1 Coordinates .................................................................................................... 7-1 Traverse ........................................................................................................... 8-1 Global Positioning System ..........................................................
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...Contents 1. 2. 3. 4. Understanding the Time Domain, Frequency Domain, and FFT Windowing Summary Additional Instrumentation Resources 1. Understanding the Time Domain, Frequency Domain, and FFT The Fourier transform can be powerful in understanding everyday signals and troubleshooting errors in signals. Although the Fourier transform is a complicated mathematical function, it isn’t a complicated concept to understand and relate to your measured signals. Essentially, it takes a signal and breaks it down into sine waves of different amplitudes and frequencies. Let’s take a deeper look at what this means and why it is useful. All Signal Are the Sum of Sines When looking at real-world signals, you usually view them as a voltage changing over time. This is referred to as the time domain. Fourier’s theorem states that any waveform in the time domain can be represented by the weighted sum of sines and cosines. For example, take two sine waves, where one is three times as fast as the other—or the frequency is 1/3 the first signal. When you add them, you can see you get a different signal. Figure 1. When you add two signals, you get a new signal. Now imagine if that second wave was also 1/3 the amplitude. This time, just the peaks are affected. 1/11 www.ni.com Figure 2. Adjusting the amplitude when adding signals affects the peaks. Imagine you added a third signal that was 1/5 the amplitude and frequency of the original signal. If you continued...
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...Preparing for College Physics David Murdock TTU October 11, 2000 2 Preface! To the Instructors: This booklet is free. You may download, copy and distribute it as you wish. . . if you find it to be of any value. It can be gotten from the URL: http://iweb.tntech.edu/murdock/books/PreSci.pdf so you will need the (equally free) Adobe Acrobat Reader to view it and print it. This piece of software is on many institutional computer systems, and if you don’t have it on your machine, go get it at http://w1000.mv.us.adobe.com/products/acrobat/readstep2.html The pages are in a format which looks best when printed double-sided. I’m giving it away to anyone in the same situation as me: You have many students in your introductory science courses who don’t have adequate preparation in basic mathematics, and you want to give them something simple and friendly to read. Preferably something that gets right to the point and which costs no more than the paper it’s printed on. I didn’t know where I could get a document like this, so I wrote one. You’ll notice that “significant figures” have not been rigidly observed in the numerical examples. That’s because this book is directed at students who need help in getting any correct numbers to round off. If you find this booklet to be useful or else worth exactly what it costs and/or have any suggestions, please write to me at murdock@tntech.edu To the Students: Your college science courses may very well require you to do some mathematics (algebra...
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...Name: _______________________________ NORTHERN BURLINGTON COUNTY REGIONAL HIGH SCHOOL Mathematics: Honors Pre-Calculus 2015 Summer Assignment Required Materials Additional Resources Graphing calculator (TI-84 or equivalent) Online resources such as: Kahn Academy www.ixl.com youtube videos www.virtualnerd.com Teacher Contact Information Due Date(s) First day of the new school year Expected time for completion jnewman@nburlington.com mdodd@nburlington.com Administrative Contact Information Approximately 6 hours Note: The time required to complete this assignment will vary. This is only a rough estimate to help with planning. Students should keep in mind their own study habits and pace of work while budgeting time to complete this assignment. Ms. Amy Stella, Director of Instruction for Mathematics astella@nburlington.com (609) 298-3900 ext 2220 Purpose of assignment: Refine and practice skills learned in prior math courses to prepare for the Honors Precalculus mathematics curriculum. Preview style of benchmark assessments and evaluations that are administered throughout the year. Skills/Knowledge required for completion: This assignment reviews some of the basic concepts of solving all types of equations, functions, simplifying radicals, unit circle, trigonometry, systems of equations Grading: Completion of this assignment will count as the first practice grade for the school year (practice is worth 10% of each marking...
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...E663 Laboratory session 8: AC Operational Amplifiers Agrim Ganti University College London 5th December 2008 ABSTRACT The report is written on an investigation which comprises of testing three operational amplifier circuits with AC signals. The three types of circuit include the integrator, the AC inverting amplifier and the AC non-inverting amplifier circuit. The integrator circuit was tested with a square-wave and a sinusoidal wave input signal at 1kHz frequency. The results showed that the square-wave input signal produced a triangular wave output whereas the sinusoidal input produced a sinusoidal output signal with a positive 90 degree phase shift. Both output signals were showed to be the integral of their relative input signals. The output voltage gain of the AC inverting and the AC non-inverting amplifier circuits were tested with a frequency range of 100Hz to 10kHz. The results were plotted on a logarithmic scaled graph which showed that both amplifiers acted like high-pass filters, each amplifier achieving its maximum gain set by the specification at higher frequencies nearer to 10kHz. The only difference between the two op-amps was that the AC inverting amplifier achieved negative gain in comparison to the positive gain achieved by the AC non-inverting amplifier. For further investigation, the frequency was increased above 10kHz for the AC inverting amplifier circuit which showed a linear fall in gain, which was explained by the theory of slew rate limitation...
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