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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…...

...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: cos15= 2+32 S: cos15=cos302 cos302= 1+cos302 cos15=1+322 cos15= 2+34 References: 4 : Young, Algebra and Trigonometry, Second Edition, Chapter7: Analytic Trigonometry p668. 5 : Young, Algebra and Trigonometry, Second Edition, Chapter7: Analytic Trigonometry p686. 6 : Young, Algebra and Trigonometry, Second Edition, Chapter7: Analytic Trigonometry p704. 7.) Use quotient identity to find tanx and cotx if sinx=35 and cosx= -45 A: tanx=-34 and cotx=......

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...Browse * Saved Papers ------------------------------------------------- Top of Form Bottom of Form * Home Page » * Other Topics History of Indian Mathematics In: Other Topics History of Indian Mathematics MATHEMATICS IN INDIA The history of maths in india is very great & eventful.Indians gave the system of numerals, zero, geometry & equations to the world. The great Indian mathematician Aryabhata (476-529) wrote the Aryabhatiya ─ a volume of 121 verses. Apart from discussing astronomy, he laid down procedures of arithmetic, geometry, algebra and trigonometry. He calculated the value of Pi at 3.1416 and covered subjects like numerical squares and cube roots. Aryabhata is credited with the emergence of trigonometry through sine functions. Around the beginning of the fifteenth century Madhava (1350-1425) developed his own system of calculus based on his knowledge of trigonometry. He was an untutored mathematician from Kerala, and preceded Newton and Liebnitz by a century. The twentieth-century genius Srinivas Ramanujan (1887-1920) developed a formula for partitioning any natural number, expressing an integer as the sum of squares, cubes, or higher power of a few integers. Origin of Zero and the Decimal System The zero was known to the ancient Indians and most probably the knowledge of it spread from India to other cultures. Brahmagupta (598-668),who had worked on mathematics and astronomy, was the head of the astronomy observatory in......

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...Trigonometry Topic Assessment 1. Solve these equations for [pic] a. [pic] b. [pic] c. [pic] [6] 2. Solve these equations for [pic]. Give your answers as a multiple of [pic]. a. [pic] b. [pic] c. [pic] [6] 3. Solve these equations for [pic] a. [pic] b. [pic] c. [pic] [9] 4. Solve these equations for [pic]. Give your answers as a multiple of [pic]. a. [pic] b. [pic] c. [pic] [9] 5. Solve these equations for [pic] a. [pic] b. [pic] c. [pic] [9] 6. Solve these equations for [pic] a. [pic] [4] b. [pic] [4] c. [pic] [3] Total 50 marks Topic Assessment Solutions 1. (i) [pic] Solutions are in the 1st and 4th quadrants. [pic] or [pic] [pic] (ii) [pic] Solutions are in the 3rd and 4th quadrants [pic] or [pic] [pic] (iii) [pic] Solutions are in 1st and 3rd quadrants. [pic] or [pic] [pic] 2. (i) [pic] Solutions are in 1st and 4th quadrants. [pic] or [pic] [pic] (ii) [pic] Solutions are in 1st and 2nd quadrants [pic] or [pic] [pic] (iii) [pic] Solutions are in 1st and 3rd quadrants [pic] or [pic] [pic] 3. (i) [pic] Solutions are in 1st and 2nd quadrants [pic] or [pic] [pic] ...

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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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...Right Triangles Word Problems with Illustrations and Solutions 1. Solve the right triangle ABC given that c = 18 cm and b = 9 cm. To find the remaining side a, use the Pythagorean Theorem: a2 + b2 = c2 a2 = c2 - b2 a2 = (18cm)2-(9cm)2 a2 = 324cm2 – 81cm2 a2 = 243cm2 a = 15.59cm 2. Ben and Emma are out flying a kite. Emma can see that the kite string she is holding is making a 70° angle with the ground. The kite is directly above Ben, who is standing 50 feet away. To the nearest foot, how many feet of string has Emma let out? cos(70°) = 50 x xcos70° = 50 x = ___50___ cos(70°) x = ___50___ .3420 x = 146.1989 3. and 4. A ladder must reach the top of a building. The base of the ladder will be 25′ from the base of the building. The angle of elevation from the base of the ladder to the top of the building is 64°. Find the height of the building (h) and the length of the ladder (m). tan64° = _h_ 25′ 2.05 = _h_ 25′ h = (25′) (2.05) h = 51.25′ cos64° = 25′ m .4384 = 25′ m m = _25′_ .4384 m = 57. 02′ Prepared by Danica Jane Maling IV-St. Ambrose (IHMA) Page 1 Right Triangles Word Problems with Illustrations and Solutions 5. The earth, moon, and sun create a right triangle during the first quarter moon. The distance from the earth to the moon is about 189,589.27 miles with an angle of 82.750. What is the distance between the sun and the moon? Solution: Let d = the distance between the sun and the moon. We can use the tangent function to find the value of d: tan(82.750)......

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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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...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......

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...MA131 0 : Module 2 Exponential a nd Logarithmic Functions Exercise 2 .2 Solving Exponential and Logarithmic Equations 1 Answer the following questions to complete this exercise: 1. Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents: 6 x = 216 2. Solve the following exponential equation: e x = 22.8 Express the solution in terms of natural logarithms. Then, use a calculator to obtain a decimal approximation for the solution. 3. Solve the following logarithmic equation: log 7 x = 2 Reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. 4. Solve the following logarithmic equation: log ( x + 16) = log x + log 16 Reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. 5. The population of the world has grown rapidly during the past century. As a result, heavy demands have been made on the world's resources. Exponential functions and equations are often used to model this rapid growth, and logarithms are used to model slower growth. The formula 0.0547 16.6 t Ae models the population of a US state, A , in millions, t years after 2000. a. What was the population in 2000? b. When will the population of the state reach 23.3 million? 6. The goal of our financial security depends on understanding how money in savings accounts grows in......

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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......

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...Variation, Joint Variation, Combined Variation Guided Learning / Working through Examples PROJECT LONG QUIZ 4 11 SUMMATIVE ASSESSMENT FINAL EXAMINATION CO1, CO2, CO3, CO4 10. Opportunities to Develop Lifelong Learning Skill To help students understand and apply the mathematical principles of Algebra and provide them with the needed working knowledge of the different mathematical concepts and methods for them to fully understand the relationship of Algebra with the increasingly complex world. 11. Contribution of Course to Meeting the Professional Component Engineering Topics : 0 % General Education : 0 % Basic Sciences and Mathematics : 100% 12. Textbook: College Algebra and Trigonometry, 7th edition Richard N. Aufmann, Vernon C. Barker, Richard D. Nation 13. Course Evaluation Student performance will be rated based on the following: Assessment Tasks Weight (%) Minimum Average for Satisfactory Performance (%) CO 1 Long Quiz 1 12.5 12.25 CPR 1 2.5 CA 1 2.5 CO 2 Long Quiz 2 12.5 12.25 CPR 2 2.5 CA 2 2.5 CO 3 Long Quiz 3 (online) 3.75 12.25 Long Quiz 3 (written) 8.75 CPR 3 2.5 CA 3 2.5 CO 4 Long Quiz 4 12.5 12.25 CPR 4 2.5 CA 4 2.5 PROJECT 5.0 3.5 Summative Assessment: Final Examination 25 17.5 TOTAL 100 70 The final grades will correspond to the weighted average scores shown below: Final Average Final Grade 96 ......

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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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...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......

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...Essential Mathematics 1: Algebra and Trigonometry Assignment One Question 1a) Solve 2+x=3x+2x-3. Leave solutions in simplest rational form. The linear equation which is in the form ax+b=0 or can be transformed into an equivalent equation into this form. 2+x=3x+2x-3 Expand 2x-3. 2+x=3x+2x-6 Add 3x+2x together. 2+x=5x-6 Subtract 5x from both sides. 2-4x=-6 Subract 2 from both sides -4x=-8 Divide both sides by-4 x=2 Check Solution x=2 2+x=3x+2x-3 Change x to 2 2+2=3×2+22-3 Add 2+2 together, multiply 3 and 2, expand 22-3 4=6+4-6 Subtract 6 from 6+4 4=4 Thus, the solution for 2+x=3x+2x-3 is x=2 . Question 1b) Solve 2xx-1+3x=x-9xx-1 This equation is rational, it can be written as the quotient of two polynomials. In addiction of expressions with unequal denominators, the result is written in lowest terms and expressions are built to higher terms using the lowest common denominator. 2xx-1+3x=x-9xx-1 multiply x-1 from 3x 2+3x-1=x-9 Expand 3x-1 2+3x-3=x-9 Subtract 3 from 2 3x-1=x-9 Subtract x from both sides 2x-1=-9 Add 1 from both sides 2x=-8 Divide 2 from both sides ...

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...Math 2 Unit 6 ‐ Similarity & Right Triangle Trigonometry Thinking Through the Lesson Protocol (TTLP) Template Concepts / Key Question Task / Activity Page Describing the essential features of a dilation 6.1 Photocopy Faux What features of a dilation are important and how can I identify these? Pas 5 Examining proportional relationships in triangles that are known to be similar to 6.2 Triangle Dilations each other based on dilations How can I describe the relationships between the sides of two triangles when one is a dilation of the other? 10 Comparing definitions of similarity based on dilations and relationships between 6.3 Similar Triangles corresponding sides and angles and Other Figures What statements can I prove about similar polygons? What other criteria can I use to determine if triangles are similar? 17 Examining proportional relationships of segments when two transversals intersect ......

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...c 2012 Math Medics LLC. All rights reserved. TRIGONOMETRIC IDENTITIES • Reciprocal identities 1 1 sin u = cos u = csc u sec u 1 1 tan u = cot u = cot u tan u 1 1 csc u = sec u = sin u cos u • Pythagorean Identities sin2 u + cos2 u = 1 1 + tan2 u = sec2 u 1 + cot2 u = csc2 u • Quotient Identities sin u cos u tan u = cot u = cos u sin u • Co-Function Identities π π sin( − u) = cos u cos( − u) = sin u 2 2 tan( csc( π π − u) = cot u cot( − u) = tan u 2 2 sec( π − u) = csc u 2 • Sum-to-Product Formulas sin u + sin v = 2 sin u+v 2 u+v 2 u+v 2 u+v 2 cos u−v 2 u−v 2 u−v 2 u−v 2 • Power-Reducing/Half Angle Formulas sin2 u = 1 − cos(2u) 2 1 + cos(2u) cos2 u = 2 1 − cos(2u) tan2 u = 1 + cos(2u) sin u − sin v = 2 cos sin cos u + cos v = 2 cos cos cos u − cos v = −2 sin sin π − u) = sec u 2 • Product-to-Sum Formulas sin u sin v = cos u cos v = sin u cos v = cos u sin v = 1 [cos(u − v) − cos(u + v)] 2 1 [cos(u − v) + cos(u + v)] 2 1 [sin(u + v) + sin(u − v)] 2 1 [sin(u + v) − sin(u − v)] 2 • Parity Identities (Even & Odd) sin(−u) = − sin u cos(−u) = cos u tan(−u) = − tan u cot(−u) = − cot u csc(−u) = − csc u sec(−u) = sec u • Sum & Diﬀerence Formulas sin(u ± v) = sin u cos v ± cos u sin v cos(u ± v) = cos u cos v sin u sin v tan u ± tan v tan(u ± v) = 1 tan u tan v • Double Angle Formulas sin(2u) = 2 sin u cos u cos(2u) = cos2 u − sin2 u = 2 cos2 u − 1 = 1 − 2 sin2 u 2 tan u tan(2u) = 1 − tan2 u ...

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