,θ = 60 degrees. So w = 1cis60.
|
|
- Amice Byrd
- 5 years ago
- Views:
Transcription
1 Proving Trig Identities with Complex Numbers Introduction Complex numbers are numbers in the form a+bi, where i = 1. They can be expressed in the form r(cosθ +isinθ) for appropriate r,θ. THis is abbreviated as rcisθ, and it is helpful to know that cisa cisb = cis(a+b). It is assumed that the reader knows the definitions of sin,cos,tan and their inverses. Example 1: Convert w = 1 2 +i 3 2 into polar form. Solution: We see that in this case, a = 1 2,b = 3 2. Then a2 +b 2 = 1, so r = 1. Because cos60 = 1 2,θ = 60 degrees. So w = 1cis60. Example 2: Convert x = 2+2i 3 into polar form. Note that x = 4w from our previous example. Then all we need is to multiply r by 4. So x = 4cis60. Hopefully any confusion regarding complex numbers and polar form is cleared.
2 Part 1: A formula for sin(x+y),sin(x y). Let a = cosx,b = sinx,c = cosy,d = siny. Then cisx cisy = cis(x+y). This also means (a+bi)(c+di) = cos(x+y)+isin(x+y). Then if we compare the imaginary parts on each side, bc+ad = sin(x+y). This means sin(x+y) = cosxsiny +sinxcosy. If we plug in y for y and use the facts that cosy = cos( y), siny = sin( y) then it can be seen that sin(x y) = bc ad = sinxcosy sinycosx. Part 2: A formula for cos(x+y),cos(x y). Remember that (a+bi)(c+di) = cos(x+y)+isin(x+y) from last part. Then if we compare the real coefficients of each side, we get ac bd = cos(x+y). Then we have cos(x+y) = cosxcosy sinxsiny. If we plug in y for y, we get cos(x y) = cosxcosy +sinxsiny. Try some simple values of x and y to convince you that these identities are valid. Example 1: Find sin2x and cos2x in terms of sinx,cosx. Solution: sin2x = sin(x+x) = sinxcosx+sinxcosx = 2sinxcosx. Meanwhile, cos(x+x) = cosxcosx sinxsinx = cos 2 x sin 2 x. These are known as the DOUBLE ANGLE FORMULAS. They are handy to know, but you can easily derive them whenever you want. Extra info: The identity cisx cisy = cis(x y) holds. If we didn t want to derive formulas for cos(x y),sin(x y) by substituting y for y, we could use complex numbers again. The only difference would be that we would have (a+bi) (c+di) = cos(x y)+isin(x y).
3 Part 3: More sines and cosines Do you notice anything if you add sin(x+y)+sin(x y)? You should end up with 2sinxcosy. If you add cos(x+y)+cos(x y) you get 2cosxcosy. These are known as the SUM-TO-PRODUCT identities. They aren t used much, but it s good to know them. Example: Find 2cos37.5cos7.5. Solution: We recognize that this equals 2cos45cos30. This is easy to evaluate: It is 6 4. It s much easier than finding cosine of 37.5 and 7.5 Part 4: DeMoivre s and applications with trig DeMoivre s Theorem tells us that (rcisθ) n = r n cis(nθ). It can be proved by induction on n for integers. (We ll only focus on integer n here). Anyway, this is very helpful with trig. You may recall that cos2x = cos 2 x sin 2 x,sin2x = 2sinxcosx. We can use DeMoivre s on this: (cosx+isinx) 2 = cos 2 x+2cosxsinxi sin 2 x = cos2x+isin2x. Comparing real and imaginary parts, we get that cos 2 x sin 2 x = cos2x,sin2x = 2sinxcosx. The reason this is helpful is that it goes beyond 2x. We can find sin3x,cos3x. (cosx+isinx) 3 = cos 3 x 3cosxsin 2 x+3cos 2 xsinxi isin 3 x = cos3x+isin3x. Then we compare real and imaginary parts: cos3x = cos 3 x 3cosxsin 2 x,sin3x = 3cos 2 xsinx sin 3 x Example: Find cos60 in terms of a,b if a = cos15,b = sin15. Solution: We can find that cos4x = cos 2 2x sin 2 2x = 1 2sin 2 2x. Now cos4x = 1 8cos 2 xsin 2 x. Then cos60 = 1 8a 2 b 2.
4 Part 5: Tangent How do we find a formula for tan(x+y)? We use the definition of tangent: tanx = sinx sin(x+y) cosx. Now tan(x+y) = cos(x+y) We can use the formulas we already have for sin(x+y), cos(x+y). Then tan(x+y) = sinxcosy+sinycosx cosxcosy sinxsiny. If we divide the numerator and denominator by cosxcosy then we get tan(x+y) = tanx+tany 1 tanxtany. If you are curious, we also have tan(x y) = tanx tany 1+tanxtany. Example: Find tan(x+y) if tanx = 2,tany = 3. Solution: This is straightforward and we should get tan(x+y) = 1. How does this relate to complex numbers? Remember that if cisx = a+bi then tanx = b a. (You can envision this, it s the definition of tangent). Now if there are two complex numbers w = 1+2i,z = 1+3i and they have arguments x,y then tanx = 2,tany = 3. Now we multiply w,z. We get wz = 1+2i+3i 6 = 5+5i. If this new complex number has argument N, then N = x+y. This results from the fact that when you multiply two complex numbers you add their angles. Now tann can be evaluated to be 1. (remember, tanx = b a. ) Part 5.5 : Arctangent The arctangent function is the inverse of tangent. That means that arctan(tan x) = x. We will denote arctangent as A(x) in this article. (example: A(1) = 45 degrees). The arctangent function has the nicest form among the arcsine, arccosine, and arctangent functions and it comes up in problems a lot. For example: Find A(2)+A(1). First we need a formula for A(x)+A(y). How do we do that? Note that A(tan(A(x)+A(y))) = A(x)+A(y) because the A(x)s and tangents cancel. Then we simplfiy with the tangent formula: tan(a(x)+a(y)) = x+y x+y 1 xy. This is because once again, A(tanx) = x. Now A(x)+A(y) = A( 1 xy ). Now we can do the problem: A(2)+A(1) = A( 3).
5 Part 6: Mega Mega Problems List (not that big) 1. Find sin 82.5 degrees. 2. Find formulas for sin( x 2 ),cos(x 2 ). 3. Evaluate 16cosxsinxcos2xcos4xcos8x. 4. Find a formula for A(x)+A(y)+A(z). 5. Evaluate cos(105) cos( 15) sin 105 sin( 15). 6. You guys will love this problem :). Let w and z be complex numbers with θ equal to the argument of w z z. Then the maximum value of tan 2 θ can be written in the form p q where p,q are relatively prime positive integers. Find p+q. Conclusion: This article covers the basic trig identities. Complex numbers provide such an easy way to derive and re-derive these identities. We hope you enjoyed it!
MATH 15a: Linear Algebra Exam 1, Solutions
MATH 5a: Linear Algebra Exam, Solutions. Let T : R 3 R 4 be the linear transformation with T( e ) = 2 e + e 2 + 3 e 3 4 e 4, T( e 2 ) = e e 2 +2 e 3 +6 e 4, and T( e 3 ) = 4 e e 2 +7 e 3 +8 e 4. (a) (6
More informationFamous IDs: Sum-Angle Identities
07 notes Famous IDs: Sum-Angle Identities Main Idea We continue to expand the list of very famous trigonometric identities, and to practice our proving skills. We now prove the second most famous/most
More informationThe goal of today is to determine what u-substitution to use for trigonometric integrals. The most common substitutions are the following:
Trigonometric Integrals The goal of today is to determine what u-substitution to use for trigonometric integrals. The most common substitutions are the following: Substitution u sinx u cosx u tanx u secx
More informationSum-to-Product and Product-to-Sum Formulas
Sum-to-Product and Product-to-Sum Formulas By: OpenStaxCollege The UCLA marching band (credit: Eric Chan, Flickr). A band marches down the field creating an amazing sound that bolsters the crowd. That
More informationsecθ 1 cosθ The pythagorean identities can also be expressed as radicals
Basic Identities Section Objectives: Students will know how to use fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions. We use trig. identities
More informationReview Problems for Test 1
Review Problems for Test Math 6-03/06 9 9/0 007 These problems are provided to help you study The presence of a problem on this handout does not imply that there will be a similar problem on the test And
More informationMTH 112: Elementary Functions
1/19 MTH 11: Elementary Functions Section 6.6 6.6:Inverse Trigonometric functions /19 Inverse Trig functions 1 1 functions satisfy the horizontal line test: Any horizontal line crosses the graph of a 1
More informationsin(y + z) sin y + sin z (y + z)sin23 = y sin 23 + z sin 23 [sin x][sinx] = [sinx] 2 = sin 2 x sin x 2 = sin ( x 2) = sin(x x) ...
Sec. 01 notes Some Observations Not Linear Doubling an angles does NOT usually double the ratio. Compare sin10 with sin 20, for example. generally,... sin(2x) 2 sinx sin IS NOT a number If x, y and z are
More informationTrigonometry Tricks. a. Sin θ= ऱम ब / णण, cosec θ = णण / ऱम ब b. cos θ= आध र / णण, sec θ= णण / आध र c. tan θ = ऱम ब / आध र, cot θ = आध र/ ऱम ब
Trigonometry Tricks 1. क स भ सम ण (Right angle) लऱय स त र (formula) णण2 = ऱम ब2 + आध र2 2. अब य द रख य LAL/KKA, (ऱ ऱ/ क ) L- ऱम ब, A- आध र, K- णण 3. अब इन क रम sin θ, cos θ, tan θ, तथ cot θ, sec θ, cosec
More informationJune 9 Math 1113 sec 002 Summer 2014
June 9 Math 1113 sec 002 Summer 2014 Section 6.5: Inverse Trigonometric Functions Definition: (Inverse Sine) For x in the interval [ 1, 1] the inverse sine of x is denoted by either and is defined by the
More informationTrigonometric Identities
Trigonometric Identities Bradley Hughes Larry Ottman Lori Jordan Mara Landers Andrea Hayes Brenda Meery Art Fortgang Say Thanks to the Authors Click http://www.ck1.org/saythanks (No sign in required) To
More informationInverse Trig Functions
6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)
More informationMTH 112: Elementary Functions
MTH 11: Elementary Functions F. Patricia Medina Department of Mathematics. Oregon State University Section 6.6 Inverse Trig functions 1 1 functions satisfy the horizontal line test: Any horizontal line
More information16 Inverse Trigonometric Functions
6 Inverse Trigonometric Functions Concepts: Restricting the Domain of the Trigonometric Functions The Inverse Sine Function The Inverse Cosine Function The Inverse Tangent Function Using the Inverse Trigonometric
More informationCK- 12 Algebra II with Trigonometry Concepts 1
14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, 3. 3. 5 9,,,,,,, 4 4 4 4 3 5 3, and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1 4.ans-1401-01 5.
More informationNext, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.
Section 6.3 - Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve:
More informationSolved Examples. (Highest power of x in numerator and denominator is ½. Dividing numerator and denominator by x)
Solved Examples Example 1: (i) (ii) lim x (x 4 + 2x 3 +3) / (2x 4 -x+2) lim x x ( (x+c)- x) (iii) lim n (1-2+3-4+...(2n-1)-2n)/ (n 2 +1) (iv) lim x 0 ((1+x) 5-1)/3x+5x 2 (v) lim x 2 ( (x+7)-3 (2x-3))/((x+6)
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More information4.3 Inverse Trigonometric Properties
www.ck1.org Chapter. Inverse Trigonometric Functions. Inverse Trigonometric Properties Learning Objectives Relate the concept of inverse functions to trigonometric functions. Reduce the composite function
More informationLIMITS, AND WHAT THEY HAVE TO DO WITH CONTINUOUS FUNCTIONS
1.3/27/13 LIMITS, AND WHAT THEY HAVE TO DO WITH CONTINUOUS FUNCTIONS Probably the hardest thing to understand and to remember, about limits, is that the limit of a function at a point has in general no
More informationChapter 7, Continued
Math 150, Fall 008, c Benjamin Aurispa Chapter 7, Continued 7.3 Double-Angle, Half-Angle, and Product-Sum Formulas Double-Angle Formulas Formula for Sine: Formulas for Cosine: Formula for Tangent: sin
More informationAll In One Multi-Color Printing A way to 100% PLUS ONE MATHEMATICS REFERENCE BOOK-CHAPTER 1 f(x) = 5x + For CBSE Schools/ State Boards with NCERT Syllabus Theory explained with simple examples NCERT Exercises
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER /2018
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER 2 2017/2018 DR. ANTHONY BROWN 2. Complex Numbers 2.1. Introduction to Complex Numbers. The first thing that it is important
More informationCore Mathematics 3 Differentiation
http://kumarmaths.weebly.com/ Core Mathematics Differentiation C differentiation Page Differentiation C Specifications. By the end of this unit you should be able to : Use chain rule to find the derivative
More informationPRE-CALCULUS TRIG APPLICATIONS UNIT Simplifying Trigonometric Expressions
What is an Identity? PRE-CALCULUS TRIG APPLICATIONS UNIT Simplifying Trigonometric Expressions What is it used for? The Reciprocal Identities: sin θ = cos θ = tan θ = csc θ = sec θ = ctn θ = The Quotient
More informationy= sin3 x+sin6x x 1 1 cos(2x + 4 ) = cos x + 2 = C(x) (M2) Therefore, C(x) is periodic with period 2.
. (a).5 0.5 y sin x+sin6x 0.5.5 (A) (C) (b) Period (C) []. (a) y x 0 x O x Notes: Award for end points Award for a maximum of.5 Award for a local maximum of 0.5 Award for a minimum of 0.75 Award for the
More informationa. y= 5x 2 +2x 3 d. 2x+5=10 b. y= 3 x 2 c. y= 1 x 3 Learning Goal QUIZ Trigonometric Identities. OH nooooooo! Class Opener: March 17, 2015
DAY 48 March 16/17, 2015 OH nooooooo! Class Opener: Find D and R: a. y= 5x 2 +2x 3 b. y= 3 x 2 c. y= 1 x 3 +2 d. 2x+5=10 Nov 14 2:45 PM Learning Goal 5.1.-5.2. QUIZ Trigonometric Identities. Mar 13 11:56
More information13 Implicit Differentiation
- 13 Implicit Differentiation This sections highlights the difference between explicit and implicit expressions, and focuses on the differentiation of the latter, which can be a very useful tool in mathematics.
More informationComplex Numbers Introduction. Number Systems. Natural Numbers ℵ Integer Z Rational Q Real Complex C
Number Systems Natural Numbers ℵ Integer Z Rational Q R Real Complex C Number Systems Natural Numbers ℵ Integer Z Rational Q R Real Complex C The Natural Number System All whole numbers greater then zero
More informationFunction and Relation Library
1 of 7 11/6/2013 7:56 AM Function and Relation Library Trigonometric Functions: Angle Definitions Legs of A Triangle Definitions Sine Cosine Tangent Secant Cosecant Cotangent Trig functions are functions
More informationDuVal High School Summer Review Packet AP Calculus
DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and
More informationChapter 8: Trig Equations and Inverse Trig Functions
Haberman MTH Section I: The Trigonometric Functions Chapter 8: Trig Equations and Inverse Trig Functions EXAMPLE : Solve the equations below: a sin( t) b sin( t) 0 sin a Based on our experience with the
More information3.1 Fundamental Identities
www.ck.org Chapter. Trigonometric Identities and Equations. Fundamental Identities Introduction We now enter into the proof portion of trigonometry. Starting with the basic definitions of sine, cosine,
More information10.7 Trigonometric Equations and Inequalities
0.7 Trigonometric Equations and Inequalities 79 0.7 Trigonometric Equations and Inequalities In Sections 0., 0. and most recently 0., we solved some basic equations involving the trigonometric functions.
More information6.1: Verifying Trigonometric Identities Date: Pre-Calculus
6.1: Verifying Trigonometric Identities Date: Pre-Calculus Using Fundamental Identities to Verify Other Identities: To verify an identity, we show that side of the identity can be simplified so that it
More informationPractice Differentiation Math 120 Calculus I Fall 2015
. x. Hint.. (4x 9) 4x + 9. Hint. Practice Differentiation Math 0 Calculus I Fall 0 The rules of differentiation are straightforward, but knowing when to use them and in what order takes practice. Although
More informationChapter 7: Trigonometric Equations and Identities
Chapter 7: Trigonometric Equations and Identities In the last two chapters we have used basic definitions and relationships to simplify trigonometric expressions and equations. In this chapter we will
More informationComplex Numbers in Trigonometry
Complex Numbers in Trigonometry Page 1 Complex Numbers in Trigonometry Author Vincent Huang The final version- with better LaTeX, more contest problems, and some new topics. Credit to Binomial-Theorem
More informationCHALLENGE! (0) = 5. Construct a polynomial with the following behavior at x = 0:
TAYLOR SERIES Construct a polynomial with the following behavior at x = 0: CHALLENGE! P( x) = a + ax+ ax + ax + ax 2 3 4 0 1 2 3 4 P(0) = 1 P (0) = 2 P (0) = 3 P (0) = 4 P (4) (0) = 5 Sounds hard right?
More information1 Lecture 20: Implicit differentiation
Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation
More informationPre-Calc Trig ~1~ NJCTL.org. Unit Circle Class Work Find the exact value of the given expression. 7. Given the terminal point ( 3, 2 10.
Unit Circle Class Work Find the exact value of the given expression. 1. cos π 3. sin 7π 3. sec π 3. tan 5π 6 5. cot 15π 6. csc 9π 7. Given the terminal point ( 3, 10 ) find tanθ 7 7 8. Given the terminal
More informationMa 221 Homework Solutions Due Date: January 24, 2012
Ma Homewk Solutions Due Date: January, 0. pg. 3 #, 3, 6,, 5, 7 9,, 3;.3 p.5-55 #, 3, 5, 7, 0, 7, 9, (Underlined problems are handed in) In problems, and 5, determine whether the given differential equation
More informationCollege Trigonometry
College Trigonometry George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 131 George Voutsadakis (LSSU) Trigonometry January 2015 1 / 25 Outline 1 Functions
More information2.1 The derivative. Rates of change. m sec = y f (a + h) f (a)
2.1 The derivative Rates of change 1 The slope of a secant line is m sec = y f (b) f (a) = x b a and represents the average rate of change over [a, b]. Letting b = a + h, we can express the slope of the
More informationComplex Numbers and Exponentials
Complex Numbers and Exponentials Definition and Basic Operations A complexnumber is nothing morethan a point in the xy plane. The first component, x, ofthe complex number (x,y) is called its real part
More informationSection 7.3 Double Angle Identities
Section 7.3 Double Angle Identities 3 Section 7.3 Double Angle Identities Two special cases of the sum of angles identities arise often enough that we choose to state these identities separately. Identities
More information3.5 Double Angle Identities
3.5. Double Angle Identities www.ck1.org 3.5 Double Angle Identities Learning Objectives Use the double angle identities to solve other identities. Use the double angle identities to solve equations. Deriving
More information2t t dt.. So the distance is (t2 +6) 3/2
Math 8, Solutions to Review for the Final Exam Question : The distance is 5 t t + dt To work that out, integrate by parts with u t +, so that t dt du The integral is t t + dt u du u 3/ (t +) 3/ So the
More information4 Exact Equations. F x + F. dy dx = 0
Chapter 1: First Order Differential Equations 4 Exact Equations Discussion: The general solution to a first order equation has 1 arbitrary constant. If we solve for that constant, we can write the general
More informationMath Final Exam Review
Math - Final Exam Review. Find dx x + 6x +. Name: Solution: We complete the square to see if this function has a nice form. Note we have: x + 6x + (x + + dx x + 6x + dx (x + + Note that this looks a lot
More informationFor a semi-circle with radius r, its circumfrence is πr, so the radian measure of a semi-circle (a straight line) is
Radian Measure Given any circle with radius r, if θ is a central angle of the circle and s is the length of the arc sustained by θ, we define the radian measure of θ by: θ = s r For a semi-circle with
More informationTRIGONOMETRY OUTCOMES
TRIGONOMETRY OUTCOMES C10. Solve problems involving limits of trigonometric functions. C11. Apply derivatives of trigonometric functions. C12. Solve problems involving inverse trigonometric functions.
More informationChapter 6: Inverse Trig Functions
Haberman MTH Section I: The Trigonometric Functions Chapter 6: Inverse Trig Functions As we studied in MTH, the inverse of a function reverses the roles of the inputs and the outputs (For more information
More informationTrigonometric Identities and Equations
Trigonometric Identities and Equations Art Fortgang, (ArtF) Lori Jordan, (LoriJ) Say Thanks to the Authors Click http://www.ck.org/saythanks (No sign in required) To access a customizable version of this
More information106 Chapter 5 Curve Sketching. If f(x) has a local extremum at x = a and. THEOREM Fermat s Theorem f is differentiable at a, then f (a) = 0.
5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function
More informationSum and Difference Identities
Sum and Difference Identities By: OpenStaxCollege Mount McKinley, in Denali National Park, Alaska, rises 20,237 feet (6,168 m) above sea level. It is the highest peak in North America. (credit: Daniel
More informationSolving Equations. Pure Math 30: Explained! 255
Solving Equations Pure Math : Explained! www.puremath.com 55 Part One - Graphically Solving Equations Solving trigonometric equations graphically: When a question asks you to solve a system of trigonometric
More information8.3 Trigonometric Substitution
8.3 8.3 Trigonometric Substitution Three Basic Substitutions Recall the derivative formulas for the inverse trigonometric functions of sine, secant, tangent. () () (3) d d d ( sin x ) = ( tan x ) = +x
More informationSolution to Review Problems for Midterm II
Solution to Review Problems for Midterm II Midterm II: Monday, October 18 in class Topics: 31-3 (except 34) 1 Use te definition of derivative f f(x+) f(x) (x) lim 0 to find te derivative of te functions
More informationBasic Trigonometry. DSchafer05. October 5, 2005
Basic Trigonometry DSchafer05 October 5, 005 1 Fundementals 1.1 Trigonometric Functions There are three basic trigonometric functions, sine, cosine and tangent, whose definitions can be easily observed
More informationSESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)
SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first
More informationMath 3 Variable Manipulation Part 4 Polynomials B COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number:
Math 3 Variable Manipulation Part 4 Polynomials B COMPLEX NUMBERS A Complex Number is a combination of a Real Number and an Imaginary Number: 1 Examples: 1 + i 39 + 3i 0.8.i + πi + i/ A Complex Number
More informationMath Analysis Chapter 5 Notes: Analytic Trigonometric
Math Analysis Chapter 5 Notes: Analytic Trigonometric Day 9: Section 5.1-Verifying Trigonometric Identities Fundamental Trig Identities Reciprocal Identities: 1 1 1 sin u = cos u = tan u = cscu secu cot
More informationTrig Identities. or (x + y)2 = x2 + 2xy + y 2. Dr. Ken W. Smith Other examples of identities are: (x + 3)2 = x2 + 6x + 9 and
Trig Identities An identity is an equation that is true for all values of the variables. Examples of identities might be obvious results like Part 4, Trigonometry Lecture 4.8a, Trig Identities and Equations
More informationPractice Problems: Integration by Parts
Practice Problems: Integration by Parts Answers. (a) Neither term will get simpler through differentiation, so let s try some choice for u and dv, and see how it works out (we can always go back and try
More informationInverse Trig Functions
Inverse Trig Functions -7-08 If you restrict fx) = sinx to the interval π x π, the function increases: y = sin x - / / This implies that the function is one-to-one, an hence it has an inverse. The inverse
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,
More informationHomework Solutions: , plus Substitutions
Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions
More information2 Trigonometric functions
Theodore Voronov. Mathematics 1G1. Autumn 014 Trigonometric functions Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics..1
More informationTrig Practice 08 and Specimen Papers
IB Math High Level Year : Trig: Practice 08 and Spec Papers Trig Practice 08 and Specimen Papers. In triangle ABC, AB = 9 cm, AC = cm, and Bˆ is twice the size of Ĉ. Find the cosine of Ĉ.. In the diagram
More informationSET 1. (1) Solve for x: (a) e 2x = 5 3x
() Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x
More informationTangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever
More informationPartial Derivatives for Math 229. Our puropose here is to explain how one computes partial derivatives. We will not attempt
Partial Derivatives for Math 229 Our puropose here is to explain how one computes partial derivatives. We will not attempt to explain how they arise or why one would use them; that is left to other courses
More informationIntroduction Derivation General formula List of series Convergence Applications Test SERIES 4 INU0114/514 (MATHS 1)
MACLAURIN SERIES SERIES 4 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Maclaurin Series 1/ 21 Adrian Jannetta Recap: Binomial Series Recall that some functions can be rewritten as a power series
More informationNOTES 10: ANALYTIC TRIGONOMETRY
NOTES 0: ANALYTIC TRIGONOMETRY Name: Date: Period: Mrs. Nguyen s Initial: LESSON 0. USING FUNDAMENTAL TRIGONOMETRIC IDENTITIES FUNDAMENTAL TRIGONOMETRIC INDENTITIES Reciprocal Identities sin csc cos sec
More information5/8/2012: Practice final A
Math 1A: introduction to functions and calculus Oliver Knill, Spring 2012 Problem 1) TF questions (20 points) No justifications are needed. 5/8/2012: Practice final A 1) T F The quantum exponential function
More informationReview Problems for Test 2
Review Problems for Test Math 0 009 These problems are meant to help you study. The presence of a problem on this sheet does not imply that there will be a similar problem on the test. And the absence
More informationChapter 2: Differentiation
Chapter 2: Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 75 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationExample. Evaluate. 3x 2 4 x dx.
3x 2 4 x 3 + 4 dx. Solution: We need a new technique to integrate this function. Notice that if we let u x 3 + 4, and we compute the differential du of u, we get: du 3x 2 dx Going back to our integral,
More information(c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + 1) at x = 0.
Chapter 7 Challenge problems Example. (a) Find the equation of the tangent line for ln(x + ) at x = 0. (b) Find the equation of the parabola that is tangent to ln(x + ) at x = 0 (i.e. the parabola has
More informationFind all of the real numbers x that satisfy the algebraic equation:
Appendix C: Factoring Algebraic Expressions Factoring algebraic equations is the reverse of expanding algebraic expressions discussed in Appendix B. Factoring algebraic equations can be a great help when
More informationALGEBRAIC LONG DIVISION
QUESTIONS: 2014; 2c 2013; 1c ALGEBRAIC LONG DIVISION x + n ax 3 + bx 2 + cx +d Used to find factors and remainders of functions for instance 2x 3 + 9x 2 + 8x + p This process is useful for finding factors
More informationx 2 = 1 Clearly, this equation is not true for all real values of x. Nevertheless, we can solve it by taking careful steps:
Sec. 01 notes Solving Trig Equations: The Easy Ones Main Idea We are now ready to discuss the solving of trigonometric equations. Recall that, generally speaking, identities are equations which hold true
More informationWelcome to AP Calculus!!!
Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure you
More informationFAIRFIELD COUNTY MATH LEAGUE (FCML) Match 4 Round 1 Arithmetic: Basic Statistics
Match 4 Round 1 Arithmetic: Basic Statistics 1.) 4.8 (or_4 4/5 or 14/5).) 40.) 0 x 1 x...x n 1 point: The geometric mean of the numbers x1, x, xn is. What is the product of the arithmetic mean and the
More informationFormula Sheet. = 1- Zsirr' x = Zcos" x-i. cotx=-- tan x. cosx cotx=-.- SlUX. 2 tan x. log, a. 1 secx=-- cosx. 1 csc x = -.- SlUX.
Formula Sheet Reciprocal Identities: 1 csc x = -.- SlUX 1 secx=-- cosx 1 cotx=-- tan x Quotient Identities: SlUX tanx=-- cosx cosx cotx=-.- SlUX Pythagorean Identities: sin" x+ cos" x = I tan ' x + I=
More informationThe Plane of Complex Numbers
The Plane of Complex Numbers In this chapter we ll introduce the complex numbers as a plane of numbers. Each complex number will be identified by a number on a real axis and a number on an imaginary axis.
More informationDefinition 1.1 Let a and b be numbers, a smaller than b. Then the set of all numbers between a and b :
1 Week 1 Definition 1.1 Let a and b be numbers, a smaller than b. Then the set of all numbers between a and b : a and b included is denoted [a, b] a included, b excluded is denoted [a, b) a excluded, b
More information6.6 Inverse Trigonometric Functions
6.6 6.6 Inverse Trigonometric Functions We recall the following definitions from trigonometry. If we restrict the sine function, say fx) sinx, π x π then we obtain a one-to-one function. π/, /) π/ π/ Since
More informationCK-12 Trigonometry - Second Edition, Solution Key
CK-1 Trigonometry - Second Edition, Solution Key CK-1 Foundation Say Thanks to the Authors Click http://www.ck1.org/saythanks (No sign in required) www.ck1.org To access a customizable version of this
More informationMATH 105: PRACTICE PROBLEMS FOR CHAPTER 3: SPRING 2010
MATH 105: PRACTICE PROBLEMS FOR CHAPTER 3: SPRING 010 INSTRUCTOR: STEVEN MILLER (SJM1@WILLIAMS.EDU Question 1 : Compute the partial derivatives of order 1 order for: (1 f(x, y, z = e x+y cos(x sin(y. Solution:
More informationChapter 5 Notes. 5.1 Using Fundamental Identities
Chapter 5 Notes 5.1 Using Fundamental Identities 1. Simplify each expression to its lowest terms. Write the answer to part as the product of factors. (a) sin x csc x cot x ( 1+ sinσ + cosσ ) (c) 1 tanx
More information10.7 Trigonometric Equations and Inequalities
0.7 Trigonometric Equations and Inequalities 857 0.7 Trigonometric Equations and Inequalities In Sections 0., 0. and most recently 0., we solved some basic equations involving the trigonometric functions.
More informationHello Future Calculus Level One Student,
Hello Future Calculus Level One Student, This assignment must be completed and handed in on the first day of class. This assignment will serve as the main review for a test on this material. The test will
More informationMath 222, Exam I, September 17, 2002 Answers
Math, Exam I, September 7, 00 Answers I. (5 points.) (a) Evaluate (6x 5 x 4 7x + 3/x 5 + 4e x + 7 x ). Answer: (6x 5 x 4 7x + 3/x 5 + 4e x + 7 x ) = = x 6 + x 3 3 7x + 3 ln x 5x + 4ex + 7x ln 7 + C. Answer:
More informationChapter 2: Differentiation
Chapter 2: Differentiation Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 82 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationπ π π π Trigonometry Homework Booklet 1. Convert 5.3 radians to degrees. A B C D Determine the period of 15
Trigonometry Homework Booklet 1. Convert 5.3 radians to degrees. A. 0.09 B. 0.18 C. 151.83 D. 303.67. Determine the period of y = 6cos x + 8. 15 15 A. B. C. 15 D. 30 15 3. Determine the exact value of
More informationTrigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters
Trigonometry Trigonometry comes from the Greek word meaning measurement of triangles Angles are typically labeled with Greek letters α( alpha), β ( beta), θ ( theta) as well as upper case letters A,B,
More informationChapter 06: Analytic Trigonometry
Chapter 06: Analytic Trigonometry 6.1: Inverse Trigonometric Functions The Problem As you recall from our earlier work, a function can only have an inverse function if it is oneto-one. Are any of our trigonometric
More informationInverse Trigonometric Functions
Inverse Trigonometric Functions Lori Jordan, (LoriJ) Brenda Meery, (BrendaM) Say Thanks to the Authors Click http://www.ck1.org/saythanks (No sign in required) To access a customizable version of this
More information