Math 115 Spring 2014 Written Homework 10SOLUTIONS Due Friday, April 25


 Maximillian Hopkins
 2 years ago
 Views:
Transcription
1 Math 115 Spring 014 Written Homework 10SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain and range of g in interval notation. Solution: There is a point on the graph corresponding to every x value except x =. Hence, the domain of g is (, (, 5]. The range is (, 1 (1, ] (b What is g(? What is g(? What is g(4? Solution: We are looking for the yvalues corresponding to the points where x =, x = and x = 4. We see that g( =, g( is undefined (as x = is not in the domain of g, and g(4 =. (c For what values of x does (i g(x =?, (ii g(x =?, (iii g(x = 1? Solution: We start by looking at the set of points on the graph of g that intersect the horizontal lines, y =, y = and y = 1, respectively. For y = we see that y = intersects the graph at the points (1,, (, and the line interval from (, to (0,. Hence, the solutions to the equation g(x = are the set {x R x 0, x = 1 or x = }. For y =, there is only one point of intersection with the graph; (,. g(x = when x =. Hence, For y = 1, there is no point of intersection with the graph. Hence, there are no solutions to the equation g(x = 1.
2 (d Determine all of the following limits from the graph of y = g(x: x g(x x g(x x + g(x x 0 g(x x 4 g(x x 4 + g(x x g(x Solution: g(x = 1 x g(x = x x + g(x = x 0 g(x does not exist (since the limit from the right of x = 0 is 1 and the limit from the left of x = 0 is x 4 g(x = 1 g(x = x 4 + x g(x =
3 . Express the lengths a and b in the figure below in terms of θ. Solution: Use the righttriangle definitions of the trig ratios and then solve for a and b: sin θ = a 4 a = 4 sin θ and cos θ = b 4 b = 4 cos θ. A boat approaches a 0ft lighthouse whose base is at sea level. Let b be the distance between the boat and the base of the lighthouse. Let L be the distance between the boat and the top of the lighthouse. Let θ be the angle of elevation between the boat and the top of the lighthouse. (a Express b as a function of θ. Solution: To model this situation, we can use a right triangle: Using the righttriangle definitions, we know tan θ = 0 0. Thus, b = b tan θ.
4 (b Express L as a function of θ. Solution: Using the same right triangle from above, sin θ = 0 0. Thus, L = L sin θ. 4. (a In a circle of radius r, an arc of length 10 is swept out by an angle of radians. What is the exact radius of the circle? Solution: The radian was defined in such a way that the arc length, s, is equal to the radius times the angle, when the angle is in radians. i.e. Here we have that s = 10 and θ =. Thus, s = rθ. 10 = r r = 10. (b In a circle of radius r =, what is the exact length of the arc swept by an angle of 10? Solution: Before we can apply the arc length formula, we must have an angle in radians. Thus, here we must convert the angle from degree measure into radian measure. 10 ( π 180 = 10π 180 = 1π 18 = 7π. Now we have r = and θ = 7π. s = rθ = ( 7π = 7π. The length of the are subtended by the angle 10 on the circle is 7π. 5. Suppose θ is an angle in standard position (meaning it starts on the positive xaxis whose radian measure is an integer value between 0 and π. (a If the angle falls in the third quadrant, what is the radian measure θ? Solution: The angles in the third quadrant are between the angles π = and = Thus, the only integer value between these is 4. So, θ = 4. π (b If the terminal side of the angle had fallen in any other quadrant, could you still answer the question with a single value? Explain. Solution: If the terminal side had fallen in quadrant I, we would be able to answer this question because those angles fall between 0 and π = Hence, in quadrant I, θ would be equal to 1. In quadrants II and IV, there would be two possible values for θ; and in II and 5 and in IV.
5 . For each angle θ given below, when sketched in standard position, determine the quadrant in which it lies. (Hint: in some cases, it might help to determine a coterminal angle that is between 0 and π. You should justify your answer in some way  either with a brief explanation or a picture. (a θ = 7π Solution: θ = 7π = π + π. π is full revolutions and then π is less than π terminal side of θ will end up in the first quadrant. so the (b θ = 11π 4 Solution: θ = 11π 4 = π π 4 clockwise π which is more than π 4 in the third quadrant.. π is 1 full revolution clockwise an then we continue but less than π. Thus, the terminal side of θ will end up (c θ = 5.5 Solution: θ = 5.5 is between the quadrantal angles π = and π = Thus, the terminal side of θ will end up in the fourth quadrant. 7. (a Suppose θ is an angle in standard position which intersects the unit circle at the point (x, y. If y = 1, what are the possible values of cos θ? If you know that cos θ is positive, in 8 which quadrant does the angle lie? Solution: The equation of the unit circle is x + y = 1. If we plug y = 1 8 we get two possible values of x and thus two possible values of cos θ: into the equation, x + y = 1 x + ( 1 8 = 1 x = 1 x = 4 x = ± 8 The possible values are cos θ = and cos θ =. If we know that cos θ is positive, then 8 8 the angle must lie in quadrant I because both the x and the y coordinate of the corresponding point on the unit circle are positive.
6 (b Suppose sin α = 4 5 and π < α < π. Find cos α, tan α, cot α, sec α, and csc α. Solution: Since π < α < π, we know that the angle lies in quadrant III. One method would be to draw a right triangle in quadrant III with the opposite side length 4 and the hypotenuse 5. This is a 45 triangle so the adjacent side is. Then we can use the right triangle trig definitions to determine the other trig values  remembering that the angle is in the third quadrant. Another method would be to use the equation for the unit circle: x + y = 1 or the equivalent identity cos (x + sin (x = 1. By plugging in sin α = y = 4, we can then solve for 5 cos α = y = 5. Either way, we get the following solutions: cos α = 5, tan α = 4, cot α = 4, sec α = 5, csc α = 5 4 (c Suppose cos β = and β lies in quadrant II. Find sin β, tan β, cot β, sec β, and csc β. Solution: One method would be to draw a right triangle in quadrant II with the adjacent side length and the hypotenuse. This is a triangle so the opposite side is 1. Then we can use the right triangle trig definitions to determine the other trig values  remembering that the angle is in the second quadrant. Another method would be to recognize the point on the Unit Circle here. The point is in ( the second quadrant with an xcoordinate of. Thus, the point is, 1. Either way, we get the following solutions: sin β = 1, tan β = 1 = 1, cot β =, sec β =, csc β = 8. Find( the exact value of each of the following. You must justify your answer in some way. 9π (a cos Solution: 9π = 8π + π = 4π + π. Thus the angle is coterminal to π. Thus, cos ( 9π ( π = cos = 0
7 (b sin ( 4π Solution: The terminal side of 4π ends up in quadrant so it s sine value will be negative. The reference angle here is π. Using either a special right triangle or the unit circle, we find that ( 4π ( π sin = sin = ( 11π (c tan Solution: The terminal side of 11π ends up in quadrant 4 so it s tangent value will be negative. The reference angle is π. Using either a special right triangle or the unit circle, we find that (d ( csc 5π 4 ( 11π tan ( π = tan = sin π cos π = 1 = 1 Solution: The angle 5π 4 will fall in quadrant with a reference angle of θ r = π 4. Using either a special right triangle or the unit circle, we find that ( csc 5π ( π = csc = sin π = 1 1 = 4 (e ( 48π sec Solution: The angle 48π = 1π is a quadrantal angle that is coterminal to π. Thus, ( 48π 1 sec = sec(1π = cos 1π = 1 cos π = 1 1 = 1 ( (f cos 11π Solution: The angle 11π = 9π π = π π will fall in quadrant 1 with a reference angle of θ r = π. Using either a special right triangle or the unit circle, we find that ( cos 11π ( π = cos = 1
8 9. As we did with the sine function in lecture, state the properties of the cosine function. (a State the domain and range of f(x := cos x. Solution: The domain is x R. The range is f(x [ 1, 1]. (b Determine all values of x where f(x is zero (i.e. the xintercepts. Solution: The zeros occur when x = π + kπ where k is any integer. (c Determine the yintercept for the graph y = cos x. Solution: The yintercept occurs at (0, Use your knowledge of transformations to sketch the graph of at least two periods/cycles of each function. ( (a f(x := sin x + π 4 Solution: We begin with the graph of y = sin x: Next we reflect the graph over the x axis to get the graph of y = sin x: Finally, we shift the previous graph to the left π ( 4 to get the graph of y = sin x + π : 4
9 (b g(x := cos(x + 4 Solution: We begin with a graph of y = cos x: Then we shift the graph up 4 to get the graph of y = cos(x + 4:
10 (c ( h(x := cos x π + 4 Solution: We begin with a graph of y = cos x. Then we shift the graph to the right π 4 to get the graph of y = cos(x π 4 Now applying the vertical stretch cause by A = yields the graph y = cos ( x π 4. Lastly, we apply the vertical shift up. This results in the final graph y = cos ( x π 4 +.
11 Remember  there is more than correct answer for each of these graphing questions. ANY complete cycles is correct. Some of you may have used a different method on part (c and ended up with the following graph:
PreCalculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.
PreCalculus II 4.2 Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle
More informationChapter 6 Trigonometric Functions of Angles
6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationTrigonometry Chapter 3 Lecture Notes
Ch Notes Morrison Trigonometry Chapter Lecture Notes Section. Radian Measure I. Radian Measure A. Terminology When a central angle (θ) intercepts the circumference of a circle, the length of the piece
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More information4.1 Radian and Degree Measure
Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position
More informationTrigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
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:
More informationFind the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.
SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.
More information5.2 Unit Circle: Sine and Cosine Functions
Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and
More informationChapter 5 The Trigonometric Functions
P a g e 40 Chapter 5 The Trigonometric Functions Section 5.1 Angles Initial side Terminal side Standard position of an angle Positive angle Negative angle Coterminal Angles Acute angle Obtuse angle Complementary
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationy = rsin! (opp) x = z cos! (adj) sin! = y z = The Other Trig Functions
MATH 7 Right Triangle Trig Dr. Neal, WKU Previously, we have seen the right triangle formulas x = r cos and y = rsin where the hypotenuse r comes from the radius of a circle, and x is adjacent to and y
More informationAngles and Their Measure
Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two
More information4.1: Angles and Radian Measure
4.1: Angles and Radian Measure An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other is called the terminal side. The endpoint that they share is
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More informationChapter 6: Periodic Functions
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a triangle, and related to points on a circle. We noticed how the x and y values
More informationPreCalculus Review Problems Solutions
MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry PreCalculus Review Problems Solutions Problem 1. Give equations for the following lines in both pointslope and slopeintercept form. (a) The
More information6.3 Polar Coordinates
6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationTrigonometry LESSON ONE  Degrees and Radians Lesson Notes
210 180 = 7 6 Trigonometry Example 1 Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles:
More informationPOLAR COORDINATES DEFINITION OF POLAR COORDINATES
POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand
More informationRoots and Coefficients of a Quadratic Equation Summary
Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and
More information55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim
Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of
More informationPractice Problems for Exam 1 Math 140A, Summer 2014, July 2
Practice Problems for Exam 1 Math 140A, Summer 2014, July 2 Name: INSTRUCTIONS: These problems are for PRACTICE. For the practice exam, you may use your book, consult your classmates, and use any other
More informationExample 1. Example 1 Plot the points whose polar coordinates are given by
Polar Coordinates A polar coordinate system, gives the coordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationDefinition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.
Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote
More informationWho uses this? Engineers can use angles measured in radians when designing machinery used to train astronauts. (See Example 4.)
1 The Unit Circle Objectives Convert angle measures between degrees and radians. Find the values of trigonometric functions on the unit circle. Vocabulary radian unit circle California Standards Preview
More informationComplex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers
Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension
More informationGive an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179
Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.
More informationCalculus with Analytic Geometry I Exam 10 Take Home part
Calculus with Analytic Geometry I Exam 10 Take Home part Textbook, Section 47, Exercises #22, 30, 32, 38, 48, 56, 70, 76 1 # 22) Find, correct to two decimal places, the coordinates of the point on the
More informationChapter 5: Trigonometric Functions of Angles
Chapter 5: Trigonometric Functions of Angles In the previous chapters we have explored a variety of functions which could be combined to form a variety of shapes. In this discussion, one common shape has
More informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More informationEvaluating trigonometric functions
MATH 1110 0090906 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More information1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic
1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single
More informationInverse Circular Function and Trigonometric Equation
Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse
More informationVolume and Surface Area of a Sphere
Volume and Surface rea of a Sphere Reteaching 111 Math ourse, Lesson 111 The relationship between the volume of a cylinder, the volume of a cone, and the volume of a sphere is a special one. If the heights
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More information+ 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider
Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationTriangle Trigonometry and Circles
Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationSemester 2, Unit 4: Activity 21
Resources: SpringBoard PreCalculus Online Resources: PreCalculus Springboard Text Unit 4 Vocabulary: Identity Pythagorean Identity Trigonometric Identity Cofunction Identity Sum and Difference Identities
More information6.1: Angle Measure in degrees
6.1: Angle Measure in degrees How to measure angles Numbers on protractor = angle measure in degrees 1 full rotation = 360 degrees = 360 half rotation = quarter rotation = 1/8 rotation = 1 = Right angle
More information4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved.
4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch
More informationSection 3.1 Radian Measure
Section.1 Radian Measure Another way of measuring angles is with radians. This allows us to write the trigonometric functions as functions of a real number, not just degrees. A central angle is an angle
More information3. Right Triangle Trigonometry
. Right Triangle Trigonometry. Reference Angle. Radians and Degrees. Definition III: Circular Functions.4 Arc Length and Area of a Sector.5 Velocities . Reference Angle Reference Angle Reference angle
More informationTrigonometry Hard Problems
Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory.
More informationIntegration Involving Trigonometric Functions and Trigonometric Substitution
Integration Involving Trigonometric Functions and Trigonometric Substitution Dr. Philippe B. Laval Kennesaw State University September 7, 005 Abstract This handout describes techniques of integration involving
More informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationSection summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2
Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More informationa cos x + b sin x = R cos(x α)
a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this
More information2312 test 2 Fall 2010 Form B
2312 test 2 Fall 2010 Form B 1. Write the slopeintercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function
More informationTrigonometry Lesson Objectives
Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the
More informationSection 63 DoubleAngle and HalfAngle Identities
63 DoubleAngle and HalfAngle Identities 47 Section 63 DoubleAngle and HalfAngle Identities DoubleAngle Identities HalfAngle Identities This section develops another important set of identities
More informationList the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated
MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible
More informationWORKBOOK. MATH 30. PRECALCULUS MATHEMATICS.
WORKBOOK. MATH 30. PRECALCULUS MATHEMATICS. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Contributor: U.N.Iyer Department of Mathematics and Computer Science, CP 315, Bronx Community College, University
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationNational Quali cations 2015
H National Quali cations 05 X77/76/ WEDNESDAY, 0 MAY 9:00 AM 0:0 AM Mathematics Paper (NonCalculator) Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given only
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationEngineering Math II Spring 2015 Solutions for Class Activity #2
Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the xaxis. Then find
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More information6.1 Basic Right Triangle Trigonometry
6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationSolution Guide for Chapter 6: The Geometry of Right Triangles
Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab
More informationMaths Pack. For the University Certificates in Astronomy and Cosmology
Maths Pack Distance Learning Mathematics Support Pack For the University Certificates in Astronomy and Cosmology These certificate courses are for your enjoyment. However, a proper study of astronomy or
More information42 Degrees and Radians
Write each decimal degree measure in DMS form and each DMS measure in decimal degree form to the nearest thousandth. 1.11.773 First, convert 0. 773 into minutes and seconds. Next, convert 0.38' into seconds.
More informationDear Accelerated PreCalculus Student:
Dear Accelerated PreCalculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, collegepreparatory mathematics course that will also
More information2. Right Triangle Trigonometry
2. Right Triangle Trigonometry 2.1 Definition II: Right Triangle Trigonometry 2.2 Calculators and Trigonometric Functions of an Acute Angle 2.3 Solving Right Triangles 2.4 Applications 2.5 Vectors: A Geometric
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationSection 1.8 Coordinate Geometry
Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of
More informationFriday, January 29, 2016 9:15 a.m. to 12:15 p.m., only
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession
More information18.4. Errors and Percentage Change. Introduction. Prerequisites. Learning Outcomes
Errors and Percentage Change 18.4 Introduction When one variable is related to several others by a functional relationship it is possible to estimate the percentage change in that variable caused by given
More informationAlgebra 2: Themes for the Big Final Exam
Algebra : Themes for the Big Final Exam Final will cover the whole year, focusing on the big main ideas. Graphing: Overall: x and y intercepts, fct vs relation, fct vs inverse, x, y and origin symmetries,
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be realvalued functions in a single variable. A function is realvalued if the input and output are real numbers
More information5.1 The Unit Circle. Copyright Cengage Learning. All rights reserved.
5.1 The Unit Circle Copyright Cengage Learning. All rights reserved. Objectives The Unit Circle Terminal Points on the Unit Circle The Reference Number 2 The Unit Circle In this section we explore some
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More information4.3 & 4.8 Right Triangle Trigonometry. Anatomy of Right Triangles
4.3 & 4.8 Right Triangle Trigonometry Anatomy of Right Triangles The right triangle shown at the right uses lower case a, b and c for its sides with c being the hypotenuse. The sides a and b are referred
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More informationPrinciples of Math 12  Transformations Practice Exam 1
Principles of Math 2  Transformations Practice Exam www.math2.com Transformations Practice Exam Use this sheet to record your answers. NR. 2. 3. NR 2. 4. 5. 6. 7. 8. 9. 0.. 2. NR 3. 3. 4. 5. 6. 7. NR
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationPROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME  TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationObjective: Use calculator to comprehend transformations.
math111 (Bradford) Worksheet #1 Due Date: Objective: Use calculator to comprehend transformations. Here is a warm up for exploring manipulations of functions. specific formula for a function, say, Given
More informationHow to Graph Trigonometric Functions
How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle
More informationSection 59 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 59 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationPlot the following two points on a graph and draw the line that passes through those two points. Find the rise, run and slope of that line.
Objective # 6 Finding the slope of a line Material: page 117 to 121 Homework: worksheet NOTE: When we say line... we mean straight line! Slope of a line: It is a number that represents the slant of a line
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationYou can solve a right triangle if you know either of the following: Two side lengths One side length and one acute angle measure
Solving a Right Triangle A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To solve
More information