...sin2x A: True 2.) Prove the identity: tanx+cotx= secx.cscx A: sinxcosx+ cosxsinx=secx.cscx sin2x+cos2xcosxsinx=secx.cscx 1 cosxsinx=secx.cscx 1cosx . 1sinx=secx.cscx secx.cscx=secx.cscx 3.) Calculate sin 3x, depending on sin x. A: sin3x=3sinx-4sin3x S: sin3x=sin(2x+x) sin2x+x= sin2xcosx+cos2xsinx sin2x+x=2sinxcosxcosx+(1-2sin2x)sinx sin2x+x=2cos2xsinx+sinx-2sin3x sin2x+x=1-sin2x2sinx+sinx-2sin3x sin2x+x=3sinx-4sin3x References: 1: http://www.math.siu.edu/previews/109/109_Topic8.pdf 2 and 3: http://www.vitutor.com/geometry/trigonometry/identities_problems.html 4.) Use the Pythagorean Identity to find cosx, if sinx= -12 and the terminal side of x lies on quadrant III A: cosx= -32 S: sin2x+cos2x=1 (-12) 2+cos2x=1 14+cos2x=1 cos2x=34 cos2x=34 cosx=32 *note that cosine in 3rd quadrant is negative 5.) Use sum and difference identity to find the exact value of sin75 A: sin75= 6+24 S: sin75=sin45+30 sin75=sin45cos30+cos45sin30 sin75=2232+ 2212 sin75= 6+24 6.) Use a half-angle identity to find the exact value of cos15 A:...
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...MAC1114 — College Trigonometry — Project 1 Instructions: Either complete the project on separate paper or type your answers using MS Word. Label each of your problems clearly and in numerical order. Once your project is complete, save the document as a pdf file and upload your file to the Dropbox called Project 1 in Falcon Online at class.daytonastate.edu. If you are submitting a handwritten document, you must write NEATLY. If you are submitting a document using MS Word, you must use the Equation Editor correctly. Points will be deducted for work that is not neatly written or the use of incorrect symbols/notation. You need to show all of your work. Part I – Proofs Recall the following definitions from algebra regarding even and odd functions: • A function f (x) is even if f (−x) = f (x), for each x in the domain of f . • A function f (x) is odd if f (−x) = −f (x), for each x in the domain of f . Also, keep in mind for future reference that the graph of an even function is symmetric about the y-axis and the graph of an odd function is symmetric about the origin. The following shows that the given algebraic function f is an even function. In Project 2 you will need to show whether the basic trigonometric functions are even or odd. Statement: Show that f (x) = 3x4 − 2x2 + 5 is an even function. Proof: If x is any real number, then f (−x) = 3(−x)4 − 2(−x)2 + 5 = 3x4 − 2x2 + 5 = f (x) and hence f is even. Now you should prove the following in a similar manner. (1) Statement: If g(x)...
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...sin(theta) = a / c | csc(theta) = 1 / sin(theta) = c / a | cos(theta) = b / c | sec(theta) = 1 / cos(theta) = c / b | tan(theta) = sin(theta) / cos(theta) = a / b | cot(theta) = 1/ tan(theta) = b / a | sin(-x) = -sin(x) csc(-x) = -csc(x) cos(-x) = cos(x) sec(-x) = sec(x) tan(-x) = -tan(x) cot(-x) = -cot(x) sin2(x) + cos2(x) = 1 | tan2(x) + 1 = sec2(x) | cot2(x) + 1 = csc2(x) | sin(x y) = sin x cos y cos x sin y | | cos(x y) = cos x cosy sin x sin y | | tan(x y) = (tan x tan y) / (1 tan x tan y) sin(2x) = 2 sin x cos x cos(2x) = cos2(x) - sin2(x) = 2 cos2(x) - 1 = 1 - 2 sin2(x) tan(2x) = 2 tan(x) / (1 - tan2(x)) sin2(x) = 1/2 - 1/2 cos(2x) cos2(x) = 1/2 + 1/2 cos(2x) sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 ) cos x - cos y = -2 sin( (x-y)/2 ) sin( (x + y)/2 ) Trig Table of Common Angles | angle | 0 | 30 | 45 | 60 | 90 | sin2(a) | 0/4 | 1/4 | 2/4 | 3/4 | 4/4 | cos2(a) | 4/4 | 3/4 | 2/4 | 1/4 | 0/4 | tan2(a) | 0/4 | 1/3 | 2/2 | 3/1 | 4/0 | Given Triangle abc, with angles A,B,C; a is opposite to A, b oppositite B, c opposite C: a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines) c2 = a2 + b2 - 2ab cos(C)b2 = a2 + c2 - 2ac cos(B)a2 = b2 + c2 - 2bc cos(A) | | (Law of Cosines) | (a - b)/(a + b) = tan 1/2(A-B) / tan 1/2(A+B) (Law of...
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...Trigonometry (1st Grading Period) Angles and Measures • Angle – is formed by two rays with a common endpoint. • Standard Position: An angle inscribed in a circle whose center is at the origin is said to be in “standard position” if one of the sides of the angle coincides with the positive ray of the x-axis. • Stationary Ray – “Initial Side” of the angle (the x-axis, abscissa side; the one that is NOT MOVING; the starting point of every angle) • Rotating Ray – “Terminal Side” of the angle (the one that MOVES, “rotates.”) • Angle Directions – o Counter – clockwise: POSITIVE (+) o Clockwise: NEGATIVE (−) o Quadrantal Angle: the terminal side (rotating ray) is in the coordinate system, which means that all the points in the rotating ray is not located in any of the four quadrants. The measurement of the angle is divisible by 90°. It can be positive (+) or negative (−), depending on the direction of the rotating ray. Degrees and Radians • Degree/s – the measurement or location • Radian/s – the distance; unit of measurement is “rad” • Conversions: o Degree to Radian: ▪ deg × π . 180 o Radian to Degree: ▪ rad × 180. π Coterminal and Reference Angles • Coterminal Angle – differs by an integral number of revolution/s; characteristic of revolutions. It is the spiral (for more than 1 revolution) that we see in the graph of the angle. It is an angle in which the terminal side coincides...
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...Difference Between Euclidean and Spherical Trigonometry 1 Non-Euclidean geometry is geometry that is not based on the postulates of Euclidean geometry. The five postulates of Euclidean geometry are: 1. Two points determine one line segment. 2. A line segment can be extended infinitely. 3. A center and radius determine a circle. 4. All right angles are congruent. 5. Given a line and a point not on the line, there exists exactly one line containing the given point parallel to the given line. The fifth postulate is sometimes called the parallel postulate. It determines the curvature of the geometry’s space. If there is one line parallel to the given line (like in Euclidean geometry), it has no curvature. If there are at least two lines parallel to the given line, it has a negative curvature. If there are no lines parallel to the given line, it has a positive curvature. The most important non-Euclidean geometries are hyperbolic geometry and spherical geometry. Hyperbolic geometry is the geometry on a hyperbolic surface. A hyperbolic surface has a negative curvature. Thus, the fifth postulate of hyperbolic geometry is that there are at least two lines parallel to the given line through the given point. 2 Spherical geometry is the geometry on the surface of a sphere. The five postulates of spherical geometry are: 1. Two points determine one line segment, unless the points are antipodal (the endpoints of a diameter of the sphere), in which case ...
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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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...Knowledge of London One to One Examinations 1. Introduction London Taxi and Private Hire (TPH) is committed to providing a fair, open, transparent and consistent Knowledge of London examination system and to assist in meeting this aim this document provides Knowledge of London Examiners with detailed guidance for conducting one to one examinations (appearances). The following guidelines address: questions that will be asked at the various stages of appearances; assessing answers to appearance questions; and the appearance marking system. The full process for learning and testing the Knowledge of London is outlined in the TPH publication ‘Applicants for a Taxi Driver’s Licence - The Knowledge of London Examination System’. 2. Stages 3, 4 and 5 - General Only ask points within 6 miles radius of Charing Cross (All London candidates only). Answers should be based on the shortest route available, unless otherwise specified by the examiner (e.g. use of Oxford Street acceptable if shortest). Traffic is irrelevant unless specified. Using more than one bridge across the River Thames is acceptable and preferred if it is the shortest route. Road works expected to last less than 26 weeks must be ignored. Where it is apparent that road works will last longer than 26 weeks (e.g. Crossrail works at Tottenham Court Road j/w Oxford Street), a candidate would be expected to find an alternative route (and be marked accordingly) after four weeks from the commencement of works. U-turns are only acceptable...
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...Willy Ngin AMAT 452: History of Mathematics Mathematical History of China and India Since the beginning of time mathematics has been a part of history. Throughout time without mathematics we wouldn’t have been able to make fundamental advances in science, engineering, technology and much more. Although every country has different histories, cultures and lifestyles; one thing that remains the same is the universal language of Mathematics. If you go to any country in the world, mathematics will always be the same. Addition will always be addition and subtraction will always be subtraction anywhere. Some of the countries who have been able to help further our discoveries and advances in mathematics were China and India. China’s history included many different wars which led to a lot of different dynasties taking over the country. Still, ”the demands of the empire for administrative services, including surveying, taxation, and calendar making, required that many civil servants be competent in certain areas of mathematics” (Katz, 2009, p. 197). It wasn’t until 1984 when they opened the tombs that they found some of the mathematic history. “Among the books was discovered a mathematics text written on 200 bamboo strips. This work, called the Suan shu shu (Book of Numbers and Computation), is the earliest extant text of Chinese mathematics.” (Katz, 2009, p. 196). This work was created during the Han Dynasty. It consisted of different problems and their solution. Alongside...
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...SCORE: (PRELIMS) EXERCISES 4 IN TRIGONOMETRY SET A Name: ___________________________________ Course, Year & Section: ____________________ Date: ____________________________________ Instructor: ________________________________ A. Tell whether a triangle with sides of the given lengths is acute, right or obtuse. 1. 11, 11, 15 2. 6, 6, 62 3. 9, 9, 13 4. 8, 83, 16 5. 8, 14, 17 B. Tell whether each group of angles could form a triangle. Write Yes or No. 1. 39°, 35°, 116° 2. 75°, 80°, 25° 3. 30°, 60°, 90° 4. 46°, 79°, 65° 5. 102°, 50°, 48° C. Draw the right triangle whose sides have the following values, and find the six trigonometric functions of the angle A. 1. a = 4, b = 3, c = 5 2. a = 2, b = 3, c = 13 3. a = 2, b = 5, c = 13 4. a = 1, b = 1/3 5. b = 21, c = 29 D. Solve the following problems. 1. The lengths of the sides of a rectangle are 5 cm and 10 cm respectively. What will be the length of the diagonal? 2. If the length of the diagonal of a square is 22 cm, what will be the length of one of the sides? 3. An 8m long ladder is leaning against a 5m high wall. How long is the foot of the ladder from the wall? 4. A lineman who is 3m away from an electrical post is holding a 12m cable wire which is connected to the top of the post. What is the height of the electric post? 5. A man travels 15 km due north, then goes 5 km due east. How far is he...
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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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...ELECTIVES: America’s Colonial Foundations (0.5) America’s Colonial Foundations provides an introduction to the major topics of the establishment of British North America, their political economic and social structures, religious and intellectual characteristics and the transition from distant citizens of Great Britain to a new American identity. It will examine changing relationships with Native Americans, development of racial slavery as a labor source, and European cultural influences on the various colonial regions. American Literature (0.5) Throughout the course of American Literature, students will be able to encounter and experience the full span of America’s rich literary history. The course begins with the literary contributions of America’s first settlers, and explores how their faith and difficult circumstances shaped their lives and the literature through which they captured these early moments of America. The course then moves through the Age of Faith, during which the core of American Literature was shaped by a strong and foundational faith, and then into the Age of Reason, during which the world of science and modern thinking started to shape the literature of the times. The study of literature then moves into the Romantic period, and then the Realist period, both of which shaped American Literature at its core and brought about significant changes to the style, structure, and purpose of literature. The introduction of Modern literature includes the literature of...
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...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 period). There will be an assessment after a short in-class review on the student’s knowledge of the review topics (assessments are worth 40% of each marking period). Name: _____________________________________ Student Self-Assessment Summer Assignment Reflection Rate yourself on the scale from one to four for each of the following statements: Absolutely! 1 Sort of 2 Not...
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...Trigonometric Functions of any Angle When evaluating any angle θ , in standard position, whose terminal side is given by the coordinates (x,y), a reference angle is always used. Notice how a right triangle has been created. This will allow us to evaluate the six trigonometric functions of any angle. Notice the side opposite the angle θ has a length of the y value of the given coordinates. The adjacent side has a length of the x value of the coordinates. The length of the hypotenuse is given by x2 + y2 . Lets say, for the sake of argument, the length of the hypotenuse is 1 unit. This would mean the following would be true. 1 y 1 cos θ = x sec θ = x y x tan θ = cot θ = x y You must think of the sine function as giving you the y value, whereas the cosine function yields the x value. This is how we will determine whether the sine, cosine, tangent, cosecant, secant or cotangent of a given angle is a positive or negative value. sin θ = y csc θ = If the angle to be evaluated is in quadrant IV, for instance, the sine of the angle θ will be negative. The cosine of θ , in this instance, will be positive, while the tangent of the angle θ will be negative. Example Evaluate the six trigonometric functions of an angle θ , in standard position, whose terminal side has an endpoint of (-3,2). The angle with terminal side is first drawn. Remember, in order to evaluate the six trigonometric ' functions for θ , use the reference angle θ . ...
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...SPECIAL POINTS OF INTEREST: Study Habits Test Taking B Y : J A S P A R T A P B A L J A N U A R Y 2 6 , 2 0 1 6 Big Ideas The Tribune Skills Leverage Learning Welcome To Advance Functions This article will discuss several aspects that will allow you to be successful in the course. from the previous semester to get the previous year’s tests. After finishing the test review, try the test questions; if you are able to solve the test questions then you have successfully prepared for the test. Big Ideas of the Course There is a ton of content that will be taught in this course. Be sure to take notes regularly and copy all examples that are explained in the classroom. In this course you will learn things such as graphing reciprocal as well as rational functions; graphing sine, cosine, and tangent functions; proving trigonometric identities; graphing exponential growth and logarithmic functions. These ideas are barely touched in grade 11 functions but taught with detail in grade 12. Be sure to brush up on graphing before the beginning of the course because it will help you throughout the course. Study Habits Be sure to come to class everyday and on time. It can’t be stressed enough that missing class will result in failure. Get connected. If you happen to miss a class talk to your peers and keep up with homework. Make sure that you do your homework and clarify any questions with the teacher...
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...are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties are analyzed to show that the approach presented in this work is a valuable contribution to the existing literature on molecular integral calculations as well as on spherical Bessel integral calculations. [ABSTRACT FROM AUTHOR]. Plane trigonometry mostly deals with the relationships among the three sides and three angles of a triangle that lies in a plane. | A ray is...
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