...Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that p 0 < q < or 0° < q < 90° . 2 Unit circle definition For this definition q is any angle. y ( x, y ) hypotenuse opposite y 1 x q x q adjacent sin q = opposite hypotenuse adjacent cos q = hypotenuse opposite tan q = adjacent hypotenuse opposite hypotenuse sec q = adjacent adjacent cot q = opposite csc q = sin q = y =y 1 x cos q = = x 1 y tan q = x 1 y 1 sec q = x x cot q = y csc q = Facts and Properties Domain The domain is all the values of q that can be plugged into the function. sin q , q can be any angle cos q , q can be any angle 1ö æ tan q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è csc q , q ¹ n p , n = 0, ± 1, ± 2,K 1ö æ sec q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è cot q , q ¹ n p , n = 0, ± 1, ± 2,K Period The period of a function is the number, T, such that f (q + T ) = f (q ) . So, if w is a fixed number and q is any angle we have the following periods. sin ( wq ) ® cos (wq ) ® tan (wq ) ® csc (wq ) ® sec (wq ) ® cot (wq ) ® T= T T T T T 2p w 2p = w p = w 2p = w 2p = w p = w Range The range is all possible values to get out of the function. csc q ³ 1 and csc q £ -1 -1 £ sin q £ 1 -1 £ cos q £ 1 sec q ³ 1 and sec q £ -1 -¥ < tan q < ¥ -¥ < cot q < ¥ © 2005 Paul Dawkins Formulas and Identities Tangent and Cotangent Identities sin q cos q tan q = cot q = cos q sin q Reciprocal Identities 1 1 csc q = sin q = sin q csc q 1 1 sec q...
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...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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...History of Trigonometric Functions Mathematics is nearly as old as humanity itself. Since antiquity, mathematics has been a fundamental to the advances in science, engineering, and philosophy. It has evolved from simple counting, measurement and calculations, and shapes through the application of abstraction, logic, and complex trigonometric functions. Trigonometric functions are the functions of angles. They relate the angles of a triangle to the lengths of its sides. These functions are important in the study of triangles and modeling periodic phenomena, among many other applications. Most people are familiar with the trigonometric functions of sine, cosine, and tangent. The functions of sine cosine and tangent are used in a variety of ways from finding refraction angles of waves to finding the angles or sides in a triangle. Two main mathematicians that have helped improve the knowledge of trigonometric functions are Aristarchus of Samos and Hipparchus. Both of these men are from the B.C. era but both made advancements with trigonometric functions using what they had. Each of the mathematicians has impacted the world of math greatly and also has influenced society today. Aristarchus of Samos was one of the mathematicians who contributed to trigonometric functions and was born in 310 BC in Samos Greece. As an astronomer Aristarchus studied the earth and material universe beyond earth’s atmosphere. Like his predecessors Aristarchus of Samos believed in the...
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...Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that p 0 < q < or 0° < q < 90° . 2 Unit circle definition For this definition q is any angle. y ( x, y ) hypotenuse y opposite 1 q x x q adjacent opposite hypotenuse adjacent cos q = hypotenuse opposite tan q = adjacent sin q = hypotenuse opposite hypotenuse sec q = adjacent adjacent cot q = opposite csc q = y =y 1 x cos q = = x 1 y tan q = x sin q = 1 y 1 sec q = x x cot q = y csc q = Facts and Properties Domain The domain is all the values of q that can be plugged into the function. sin q , q can be any angle cos q , q can be any angle 1ö æ tan q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è csc q , q ¹ n p , n = 0, ± 1, ± 2,K 1ö æ sec q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è cot q , q ¹ n p , n = 0, ± 1, ± 2,K Range The range is all possible values to get out of the function. csc q ³ 1 and csc q £ -1 -1 £ sin q £ 1 -1 £ cos q £ 1 sec q ³ 1 and sec q £ -1 -¥ < tan q < ¥ -¥ < cot q < ¥ Period The period of a function is the number, T, such that f (q + T ) = f (q ) . So, if w is a fixed number and q is any angle we have the following periods. 2p w 2p = w p = w 2p = w 2p = w p = w sin ( wq ) ® T= cos (wq ) ® T tan (wq ) ® T csc (wq ) ® T sec (wq ) ® T cot (wq ) ® T © 2005 Paul Dawkins Formulas and Identities ...
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...How much weight did I just lift? You might ask yourself after a good set of pushups. Was it 90% of my body weight? No, maybe it was 50%? I will calculate the percentage of your body weight that you would expect to "push up" during both regular and inclined pushups. Before I begin with the math, lets define what a pushup is. More specifically, lets discuss proper form and technique. First, get onto the ground. Elevate your body using your arms and keep you back straight as a board. Don't let your gluteus maximus stick into the air or hang low. There should be a 90 degree angle between your arms and the floor. Your hands should be placed about one and a half times your shoulder width apart and pointed parallel to your body. Your body should be raised on the balls of your feet. Your feet should also be touching or no more than shoulder width apart. When you go downward, only bend your elbows. You can come back up once the elbows break the plane of your back. I will calculate the percentage of body weight resisted during a pushup for an average sized person. Since the resulting number will be a percentage, it will be correct for any person who has the same dimensions or ratio of dimensions as the average person calculated here. The characteristics of an average 25 year old American male are: Height: 70 inches (1.778 m) Palm to Shoulder length: 23 inches (0.5842 m) Shoulder to Hip Length: 24.75 inches (0.62865 m) Hip to Ankle Length: 31.5 inches (0...
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...Pg. 977 #70 5 1/5∑ai I=1 Basically what this means is that you are going to do 1/5(2.9+3.6+4.5+4.9+5.5) The 5 above the symbol means how many numbers you are going to have within the () since you need the average you multify by 1/5 (or divide by 5) a. 1/5 (2.9+3.6+4.5+4.9+5.5)= 4.28 billion b. an= .65 (1)+ 2.3 =2.95 an= .65 (2)+ 2.3=3.9 an= .65 (3)+ 2.3=4.25 an= .65 (4)+ 2.3=4.9 an= .65 (5)+ 2.3=5.55 the problem states that n= 1, 2, 3, 4, 5 so you substitute that in each other the equations then add the sums together and multiply by 1/5 21.55(1/5) =4.31 This is reasonable in terms of the actual sum. Pg.986 #62 & 70 62.) a. t(n) = [17.6 + (0.83)(n - 1)]% b. 2018 - 1967 = 51 so when the year is 2018, n = 51 and t(n) = [17.6 + (0.83)(51 - 1)]% = [17.6 + (0.83)(50)]% = (17.6 + 41.5)% = 59.1% 70.) 30 + 32 + 34 + ... + (30 + 2*25) 30 + 32 + 34 + ... + 80 26/2 * (30 + 80) = 1,430 Pg. 1000 #70 A.) 22.49 / 22.12 = -1.02 22.86 / 22.49 = -1.02 23.41 / 22.86 = -1.02 B.) the general term is of the form a_n = 22.12 * (1.02)^(n-1) for n = 1, 2, 3, ... (n = 1 represents 2003.) C.) n = 8 to get a_8 = 22.12 * (1.02)^7 = -25.41 million...
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...Theft of intellectual property at Trig Enterprises, Inc. Librium Author Note This paper is being submitted on November 19, 2015, for Kevin Harris’s ISSC351-Computer Forensics course. Trig Enterprises, Inc. is an international supplement dietary company and it is located in St. Louis. In 2006, a newly designed and ready to manufacture product was comprised. One of the Trig’s competitors released the very similar product on the market before Trig did. A short time later, it’s found that the competitor’s website was very similar to the Trig’s website. This raised a flag about the company’s 22-year old graphics designer, Kevin K. The CEO of the company, contacted the IT department to monitor his computer/laptop and email accounts, his VOIP phone and wireless devices if there was any suspicious activity. IT department scanned his emails for the last a couple of years. No questionable information found. IT department continue to monitor his devices and emails for about three months. While he was under surveillance, another similar event happened. This time a product on the manufacturing process was compromised. This caused the company millions of dollars. CEO did not a choice but the contact the local police. After contacting the local police, the police’s compu In 2006, 22-year old graphics designer, Kevin K., suspected to leak the extremely important corporate data to its competitors. The owner of the company tries to solve the issue with...
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...Therefore, you need to prepare for different types of questions in addition to those you encounter in assignments. This is especially true in view of the fact that, for instance, it is difficult in an e-assignment to ask for intermediate work in obtaining a derivative starting with its definition – namely using a limit. In the following I will attempt to give a flavour of how some questions may be worded.] There will be 4 questions of equal weight, drawn from various topics. It is entirely possible that a topic found in the text but not appearing below may appear in your test, i.e. what appears below does not constitute an exhaustive list of examinable topics. 5 possible questions are presented below. lim trig expression involving a&h Q. a) If a and b are constants, evaluate h→0 trig expression involving b&h A correct answer without detailed reasoning will be given little merit. b) Something involving related rates. Q. Obtain the derivative of the following functions, and simplify as far as possible: … functions not shown…say two different part questions involving log,...
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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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...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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...Abortion is a highly controversial issue in the United States today. There are those who argue that life beings at conception and that terminating a pregnancy is equivalent to murder. These pro-life supporters have relatively immovable opinions on abortion. In the past, they have done all they could to prevent abortions, using tactics ranging from protesting and lobbying to methods as deadly as bombing abortion clinics. On the other hand, there are pro-choice supporters as well who argue that women have the right to choose what happens to their own bodies and that a fetus is not a living person, but a part of the mother herself. They propose that though abortions should always be limited, psychological, maturity, and economic issues are all acceptable reasons for women to be able to choose whether or not they want to carry a child for nine months. In April 2009, 2008 Republican Vice-Presidential candidate Sarah Palin gave a speech at Evansville, Indiana, where she argues against abortion. She fights for her beliefs, claiming that “life is ordained, life is precious” and no selfish decisions should stand in the way of a life from living. Her ethos and utilization of pathos are both tactical techniques used to argue against the practice of abortion to try and convince her audience to fight against it as well. Palin’s entire argument against abortion would not matter unless she had any credibility that would allow the audience to deem her respectable and her opinion worth listening...
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...________________________ Sec. 6.3 (Vectors) 7. [pic][pic] is a vector in ____ form. 7. __________________ 8. If [pic] = [pic] , the magnitude of [pic] is: 8. [pic] = _____________________ and to find direction angle [pic] you use the formula: _____________________ 9. In the expression: 5 [pic] [pic], 5 is called a 9. __________________ 10. The sum of two vectors is called the 10. __________________ 11. Given [pic] = [pic] , to find the vector 11. ____________________ of one unit in the same direction as [pic] 12. [pic] and [pic] are ___ vectors. 12. __________________ 13. 2i – 3j is called a 13 ___________ _______________. 14. The trig form of a vector with 14._______________________ magnitute [pic] and direction angle [pic] is Sec. 6.4 (Vectors) 15. If [pic] = [pic] and [pic] = [pic] u [pic] v = 15._______________________ 16. If [pic] = [pic] and [pic] = [pic] to find the angle between the two vectors use: 16.________________________ (not Law of Cosines) Sec. 6.5 (Complex numbers) 17. The rectangular...
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...Contents 0. Preface 1. Functions and Models 1.1. Basic concepts of functions 1.2. Classification of functions 1.3. New functions from old functions 1 2 2 5 8 0. Preface Instructor: Jonathan WYLIE, mawylie@cityu.edu.hk Tutors: Radu Gogu, rgogu2@student.cityu.edu.hk. Texts: Single Variable Calculus, by James Stewart, 6E. In this semester, we will cover the majority of Chap 1-4, 7, 12. Upon completion of this course, you should be able to understand limit, derivatives, and its applications in mathematical modeling and infinite series. 1 2 1. Functions and Models In this chapter, we will briefly recall functions and its properties covered by high school. 1.1. Basic concepts of functions. Text Sec1.1: 5, 7, 39, 57, 67. Definition 1.1. A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. Usually, we write a function f : x → f (x) where (1) x ∈ D, i.e. x belongs to a set D , called the Domain; (2) f (x) ∈ E, i.e. f (x) belongs to a set E, called the Range; (3) x is independent variable, (4) f (x) is dependent variable. 3 For a function f , its graph is the set of points {(x, f (x)) : x ∈ D} in xy-plane. One can also use a table to represent a function. Example 1.1. Sketch the graph of following two piecewise defined functions. (1) f (x) = |x|. i.e. Absolute value of x. (2) f (x) = [x]. i.e. largest integer not greater than x. The graph of a function is a curve. But the question is: which curves are graphs of...
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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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...Goal: Students will use the trigonometric ratios (Sine, Cosine, and Tangent) to measure the real life height of various objects. How: Students will work with one partner, however they both will have to turn in individual projects. Step One Create a clinometer, by following these steps. ▪ Cut out the photocopied protractor. ▪ Glue it to your piece of cardboard. ▪ Cut out the cardboard so that it is the shape of your protractor ▪ Cut a 4-inch piece of string, tape it to the middle of the protractor by your mark. It is important that you tape your string before you tape your straw. ▪ Tape your straw to the top flat end of your protractor. ▪ At the end of the string, tape a couple pennies to it. Your clinometer should look like this: [pic] Step Two Measure the height of the wall using sine, cosine, tangent, and your clinometer. ▪ In order to measure the height of the wall, complete the Clinometer worksheet that is attached to this project sheet. ▪ After you measure the height of the wall, you will measure one more object at school, and one more at home for homework. Step Three Draw a story/cartoon that shows how you used trigonometry to solve the problem that...
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