Chapter 7. 1 a The length is a function of time, so we are looking for the value of the function when t = 2:
|
|
- Alberta Wiggins
- 6 years ago
- Views:
Transcription
1 Practice questions Solution Paper type a The length is a function of time, so we are looking for the value of the function when t = : L( ) = 0 + cos ( ) = 0 + cos ( ) = 0 + = cm We are looking for the minimum value of a function of the form y = acos [ ( x + c) ]+ d. Since the smallest value of cosine is, the minimum value of L will e: L min = 0 + ( ) = 8 cm Note: We have found the minimum value of the function using the fact that the minimum value of cosine is. We could have found the result differently: The minimum value of a function of the form y = acos [ ( x + c) ]+ d is equal to the vertical shift d minus the amplitude a ; so we have: min = d a = 0 = 8. c We have to find the least value of t such that L = 8. From part we can see that the lowest value will e otained when the cosine equals ; hence, t = t =. Note: We could find t y solving the equation: 0 + cos ( t) = 8 cos ( t) = 8 0 cos ( t ) =. t = t = sec d We have to determine the period of a function of the form y = acos [ ( x + c) ]+ d. So, the period is = = sec. Solution Paper type We will oserve the function L = 0 + cos ( t). Firstly, we have to select radian measure and then input the function. a We have to find L(): c For the remaining parts of the question, we will use the graph of the function. So, we have to set a suitale window. The minimum value is the vertical shift minus the amplitude, and the maximum value is the vertical shift plus the amplitude. Therefore, we can use y-values from (less than) 0 = 8 to (greater than) 0 + =.
2 Another way would e to use the ZoomFit feature. The window will not e suitale for all the calculations, ut we can read out the range and then just extend it a it in oth directions. We can solve oth parts and c y finding the minimum value with the smallest x. d We can find the period of a trigonometric function y finding two successive minimum points, or two successive maximum points, and then sutracting their x-coordinates. We already know the smallest positive minimum has x = 0., so we have to find the next positive minimum. Hence, we can conclude that the period is. 0. =. sin x cos x + = 0 ( cos x) cos x + = 0 Use the Pythagorean identity sin x = cos x. cos x cos x + = 0 t t + = 0 t =., t = Sustitute cos x = t. cos x =. has no solution cos x = x = 0, Solutions are: 0,. The perimeter of the shaded sector contains two radii and an arc. So, the length of the arc can e calculated: = + s s =. We can use the arc length formula to find the angle θ in radians: s = rθ = θ θ = For angles θ and θ : θ+ θ = θ =.
3 a i The amplitude of function f is, so the minimum value of the function is. ii Period of g = =. It is quickest to find the numer of solutions from the graph. We can see that there are four points of intersection; hence, there are four solutions to the equation. For d = p + q m t cos : a To find p we have to determine the mid-line; therefore, we have to find the average of the function s maximum and minimum value: + p = = To determine q, we have to determine the amplitude. The amplitude is the difference etween the function s maximum value and the mid-line: q = = 9. So, q is 9 or 9. From the given data, we can estalish that the graph starts from the maximum value, so q is positive; hence, q = 9. c From the data, we can see that the distance etween two successive maximum points (or minimum points) is 0. seconds. So, the period is 0.. Using the formula for period, we have: period = = m. Hence, m = 0.. m Note: It is useful to highlight the asic data using a rough sketch Solutions are: 0,.0,.0. Note: The question does not tell us to use a specific method to solve the equation, so we can choose any method. The most suitale method is graphical, ecause we can see the numer of solutions and find them all.
4 7 a Use the sustitution cos x = t : t + t + = 0 t =, t = cos x = has no solutions cos x = x =, Solutions are:,. Use doule angle identity for sine: - y x sinx cos x cos x = 0 Factorize: cos x = 0 x =, cosx( sin x )= 0 sin x = 0 sin x = x =, Solutions are:,,,. 8 Given < x <, it follows that sin x > 0 and cos x < 0. a Using the Pythagorean identity for sine, we have: sin cos 8 x = x = =. 9 9 Using the fact that sin x > 0, we have: sin x = =. 9 Using the doule angle identity for cosine, we have: cos cos 8 7 x = x = = = c Using cos x < 0, we have: cos x = = = 9. Hence, sin sin cos x = x x =. = 9 9 We have to interpret the data as points on a graph: High (maximum tide) at :00 with depth of.8 m ( 8,. ) (,.8) y x Low (minimum) tide at 0:0 with depth. m ( 0.,. ) a The function can e written in the form: d = Asin ( B( x + C) )+ D (or similar with the cosine function). Firstly, we will determine D, so we have to find the average of the function s maximum and minimum value: D = =. Next we will determine A. The amplitude of the function is the difference etween the function s maximum value and the mid-line: A = 8.. =.. So, A is. or.. From the data, we can see that the earlier minimum is reached at ( 0.,. ), so the graph after the mid-line first reaches the maximum value, then the minimum value; therefore, A is positive, A =.. (-0.,.) 0 (0.,.) (,.8) (0.,.) 0
5 We can also see from the data that the distance etween successive maximum and minimum points is. hours. So, half of theperiod =.. Using the formula for period, we have: period = = B =. B B To determine the horizontal translation (phase shift), we have to find the points of the graph on the mid-line. Ascissae of those points are on a distance of a of the period from the ascissa of the maximum point. Hence, ±, so 9 and. Thus, C = 9 and the function is: (.,.) (,.8) (7.7,.) 0 (0.,.) 9 d =. sin x +.. We have to find the value of the function at noon, that is, hours after midnight; therefore, d( ). sin.. 9 = + metres. c We have to find the time interval in which the value of d is greater than.. We have to convert from decimal numers to minutes: Therefore, from aout :7 pm to 7: pm the oat can dock safely. We can also see that around midnight the oat can dock safely again.
6 0 Use the sustitution tan x = t : t + t = 0 t =, t = tan x = x = tan x = cannot e solved exactly Solutions are:, 89.. a We will use the arc length formula: s = rθ = 0 = cm The angle in the shaded region is ; therefore, using the area of a sector formula: A = r = 0 θ 9 cm The function f ( x) = cos x reaches a maximum value of and a minimum value of. That means that f ( x ). The function will not reach values greater than or less than. Hence, for k > and k <, the equation will have no solutions. From the graph, we can see that the point (0, ) is on the mid-line, so k =. To determine a, we need to find the amplitude. The amplitude is the difference etween the function s maximum value and the mid-line: a = = ; so, a is or. We can see that the graph, starting from initial position at x = 0, first reaches the minimum value, and then the maximum value, so a is negative; hence, a =. We can write tan sin α cos α a using secant only: tan α = = = = sec α cos α cos α cos α So, the equation is: sec α sec α 0 = 0 sec α sec α = 0 Using the sustitution sec α = t, we have: t t = 0 t =, t =. Since α is in the second quadrant, secant should e negative and the solution is sec α =. a Using the compound angle formula for sine, we have: sin( α + β) = sin αcos β + cos α sin β From the upper triangle, we can see that: sin α = 7, cos α = 8 7 From the lower triangle, we can see that: sin β = 0, cos β = So, sin( α + β) = + = = cos( α + β) = cos αcos β sin α sin β = 8 8 = sin ( α + β) c tan( α + β) = cos ( α + β) = = = 8 Note: We can find tanα and tan β from the drawing, and then use the compound formula for tangent. From the diagram, we can see that: sin p = = and cos p = =. + + Hence: sinp = sin p sin p = =. For sinp, we can use the compound angle formula: sinp = sin p + p sinp cos p cosp sin p ( ) = +
7 So, we have to find cosp = cos p sin p = =. Finally, we have: sinp = + = =. 7 If B is otuse, then the sine is positive and cosine negative. a sin B = = + cos B = = + c sin sin cos 0 B = B B = = 9 d 9 cosb = cos B sin B = = 9 Note: In the solution, we have used the property of angles and sides in a right triangle. We can find the solution y using the Pythagorean identity for sine and cosine: sin x = sinx = cos x cos x 9 sin x + cos x = cosx cos x cos + = x = cos x = = 9 Finally, sin x = =. Now we can continue using the doule angle formulae. tan θ 8 Using the doule angle formula for tangent, we have: tan θ = = 8 tanθ = tan θ tan θ Using the sustitution tan θ = t, we will have a quadratic equation: t + 8t = 0 t =, t =. Hence, the possile values of tan θ are:,. 9 We will use the compound angle formula for sine: sinx cos α cosx sin α = k ( sinx cos α + cosx sin α) Then we will divide oth sides y cosx cos α : sinx cos α cosx sin α sinx cos α cosx sin α = k + cosx cos α cosx cos α cos x cos α cos cos α x tanx tan α = k ( tanx + tan α) tanx k tan x = k tanα + tan α tanx ( k)= tan α ( k + ) tan α( k + ) tan x = k ( k + ) Note: If we rearrange the equation differently, we otain the result tan x = tan α, which is equivalent to the k aove result. 0 We can see that either tanθ = or tanθ =. So, tan θ = θ = + k θ = + k θ =, + = tan θ = θ = + k θ = + k θ =, + = Hence, the solutions are: ±, ±. 8 8 a We have to set the window on the GDC to e the same as on the grid. 7
8 The domain of the functions is x, so we just have to draw this part of the graph. c No solutions Cosine function has a range from. But, here, its domain is restricted. Since cosine is an even function, it is enough to oserve its ehaviour on the interval from 0 to. Hence, the function is decreasing on this interval; its largest value is cos0=, and the smallest value cos. Hence, the range is the set [ cos, ]. Let CAD ˆ = θ and AC = x. From the drawing, we can see that: tan θ = x = and tan θ = = x tan θ x x tan θ. Hence: = tanθ tan θ ( tan θ) Using the doule angle formula for tangent, we have: = = = tan θ tan θ = tan θ tan θ tan θ Since θ is an angle in a right triangle, its tangent has to e positive. So, tan θ = θ.. Note: We can solve the equation = graphically. tanθ tan θ We are looking for the angle when those two sides are the same. We have to solve the equation sec x 0 = +. Hence, x x 8 sec cos = = = 8 x 8 cos = We need the first positive solution, so: x 8 8 = arccos x = arccos Hence, the solution of the equation is: 8 arccos, and the width of the water surface in the channel is: arccos = arccos cm. 8
Core Mathematics 2 Trigonometry
Core Mathematics 2 Trigonometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Trigonometry 2 1 Trigonometry Sine, cosine and tangent functions. Their graphs, symmetries and periodicity.
More informationPractice Test - Chapter 4
Find the value of x. Round to the nearest tenth, if necessary. 1. An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the length of the side opposite.
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 informationChapter 1. Functions 1.3. Trigonometric Functions
1.3 Trigonometric Functions 1 Chapter 1. Functions 1.3. Trigonometric Functions Definition. The number of radians in the central angle A CB within a circle of radius r is defined as the number of radius
More informationweebly.com/ Core Mathematics 3 Trigonometry
http://kumarmaths. weebly.com/ Core Mathematics 3 Trigonometry Core Maths 3 Trigonometry Page 1 C3 Trigonometry In C you were introduced to radian measure and had to find areas of sectors and segments.
More informationCore Mathematics 3 Trigonometry
Edexcel past paper questions Core Mathematics 3 Trigonometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Maths 3 Trigonometry Page 1 C3 Trigonometry In C you were introduced to radian measure
More informationMA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically
1 MA40S Pre-calculus UNIT C Trigonometric Identities CLASS NOTES Analyze Trigonometric Identities Graphically and Verify them Algebraically Definition Trigonometric identity Investigate 1. Using the diagram
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 informationChapter 4/5 Part 2- Trig Identities and Equations
Chapter 4/5 Part 2- Trig Identities and Equations Lesson Package MHF4U Chapter 4/5 Part 2 Outline Unit Goal: By the end of this unit, you will be able to solve trig equations and prove trig identities.
More informationChapter 4 Trigonometric Functions
Chapter 4 Trigonometric Functions Overview: 4.1 Radian and Degree Measure 4.2 Trigonometric Functions: The Unit Circle 4.3 Right Triangle Trigonometry 4.4 Trigonometric Functions of Any Angle 4.5 Graphs
More informationPractice 14. imathesis.com By Carlos Sotuyo
Practice 4 imathesis.com By Carlos Sotuyo Suggested solutions for Miscellaneous exercises 0, problems 5-0, pages 53 to 55 from Pure Mathematics, by Hugh Neil and Douglas Quailing, Cambridge University
More informationUsing this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.
Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive
More informationTrig Identities, Solving Trig Equations Answer Section
Trig Identities, Solving Trig Equations Answer Section MULTIPLE CHOICE. ANS: B PTS: REF: Knowledge and Understanding OBJ: 7. - Compound Angle Formulas. ANS: A PTS: REF: Knowledge and Understanding OBJ:
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 informationMTH 122: Section 204. Plane Trigonometry. Test 1
MTH 122: Section 204. Plane Trigonometry. Test 1 Section A: No use of calculator is allowed. Show your work and clearly identify your answer. 1. a). Complete the following table. α 0 π/6 π/4 π/3 π/2 π
More information(c) cos Arctan ( 3) ( ) PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER
PRECALCULUS ADVANCED REVIEW FOR FINAL FIRST SEMESTER Work the following on notebook paper ecept for the graphs. Do not use our calculator unless the problem tells ou to use it. Give three decimal places
More informationMATH 32 FALL 2012 FINAL EXAM - PRACTICE EXAM SOLUTIONS
MATH 3 FALL 0 FINAL EXAM - PRACTICE EXAM SOLUTIONS () You cut a slice from a circular pizza (centered at the origin) with radius 6 along radii at angles 4 and 3 with the positive horizontal axis. (a) (3
More informationHonors Algebra 2 Chapter 14 Page 1
Section. (Introduction) Graphs of Trig Functions Objectives:. To graph basic trig functions using t-bar method. A. Sine and Cosecant. y = sinθ y y y y 0 --- --- 80 --- --- 30 0 0 300 5 35 5 35 60 50 0
More informationSANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.
More information5.1: Graphing Sine and Cosine Functions
5.1: Graphing Sine and Cosine Functions Complete the table below ( we used increments of for the values of ) 4 0 sin 4 2 3 4 5 4 3 7 2 4 2 cos 1. Using the table, sketch the graph of y sin for 0 2 2. What
More informationA-Level Mathematics TRIGONOMETRY. G. David Boswell - R2S Explore 2019
A-Level Mathematics TRIGONOMETRY G. David Boswell - R2S Explore 2019 1. Graphs the functions sin kx, cos kx, tan kx, where k R; In these forms, the value of k determines the periodicity of the trig functions.
More informationThe six trigonometric functions
PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonomic Functions 4.: Trigonomic Functions of Acute Angles What you'll Learn About Right Triangle Trigonometry/ Two Famous Triangles Evaluating
More informationSection 6.1 Sinusoidal Graphs
Chapter 6: Periodic Functions In the previous chapter, the trigonometric functions were introduced as ratios of sides of a right triangle, and related to points on a circle We noticed how the x and y values
More information!"#$%&'(#)%"*#%*+"),-$.)#/*01#2-31#)(.*4%5)(*6).#* * *9)"&*#2-*5$%5%.-&*#%5)(*8).#*9%$*1*'"),-$.)#/*31#2-31#)(.*5$-51$1#)%"*(%'$.
!"#$%&'(#)%"*#%*+"),-$.)#/*0#-3#)(.*4%5)(*6).#* * 78-.-*9)"&*#-*5$%5%.-&*#%5)(*8).#*9%$**'"),-$.)#/*3#-3#)(.*5$-5$#)%"*(%'$.-:* ;)(*(%'8&**#).* )"#$%&'(#)%":*!*3*##()">**B$#-$*8$>-C*7DA*9)8-*%9*.%3-*5%..)
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
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 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 informationThese items need to be included in the notebook. Follow the order listed.
* Use the provided sheets. * This notebook should be your best written work. Quality counts in this project. Proper notation and terminology is important. We will follow the order used in class. Anyone
More informationTrigonometry LESSON SIX - Trigonometric Identities I Lesson Notes
LESSON SIX - Trigonometric Identities I Example Understanding Trigonometric Identities. a) Why are trigonometric identities considered to be a special type of trigonometric equation? Trigonometric Identities
More informationMATH 32 FALL 2013 FINAL EXAM SOLUTIONS. 1 cos( 2. is in the first quadrant, so its sine is positive. Finally, csc( π 8 ) = 2 2.
MATH FALL 01 FINAL EXAM SOLUTIONS (1) (1 points) Evalute the following (a) tan(0) Solution: tan(0) = 0. (b) csc( π 8 ) Solution: csc( π 8 ) = 1 sin( π 8 ) To find sin( π 8 ), we ll use the half angle formula:
More informationSince 1 revolution = 1 = = Since 1 revolution = 1 = =
Fry Texas A&M University Math 150 Chapter 8A Fall 2015! 207 Since 1 revolution = 1 = = Since 1 revolution = 1 = = Convert to revolutions (or back to degrees and/or radians) a) 45! = b) 120! = c) 450! =
More informationa) Draw the angle in standard position. b) determine an angle that is co-terminal to c) Determine the reference angle of
1. a) Draw the angle in standard position. b) determine an angle that is co-terminal to c) Determine the reference angle of 2. Which pair of angles are co-terminal with? a., b., c., d., 3. During a routine,
More informationSection 7.2 Addition and Subtraction Identities. In this section, we begin expanding our repertoire of trigonometric identities.
Section 7. Addition and Subtraction Identities 47 Section 7. Addition and Subtraction Identities In this section, we begin expanding our repertoire of trigonometric identities. Identities The sum and difference
More informationTO EARN ANY CREDIT, YOU MUST SHOW STEPS LEADING TO THE ANSWER
Prof. Israel N. Nwaguru MATH 11 CHAPTER,,, AND - REVIEW WORKOUT EACH PROBLEM NEATLY AND ORDERLY ON SEPARATE SHEET THEN CHOSE THE BEST ANSWER TO EARN ANY CREDIT, YOU MUST SHOW STEPS LEADING TO THE ANSWER
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 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 informationAFM Midterm Review I Fall Determine if the relation is a function. 1,6, 2. Determine the domain of the function. . x x
AFM Midterm Review I Fall 06. Determine if the relation is a function.,6,,, 5,. Determine the domain of the function 7 h ( ). 4. Sketch the graph of f 4. Sketch the graph of f 5. Sketch the graph of f
More informationD. 6. Correct to the nearest tenth, the perimeter of the shaded portion of the rectangle is:
Trigonometry PART 1 Machine Scored Answers are on the back page Full, worked out solutions can be found at MATH 0-1 PRACTICE EXAM 1. An angle in standard position θ has reference angle of 0 with sinθ
More informationPrecalculus Midterm Review
Precalculus Midterm Review Date: Time: Length of exam: 2 hours Type of questions: Multiple choice (4 choices) Number of questions: 50 Format of exam: 30 questions no calculator allowed, then 20 questions
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationUnit 3 Trigonometry Note Package. Name:
MAT40S Unit 3 Trigonometry Mr. Morris Lesson Unit 3 Trigonometry Note Package Homework 1: Converting and Arc Extra Practice Sheet 1 Length 2: Unit Circle and Angles Extra Practice Sheet 2 3: Determining
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 informationFind all solutions cos 6. Find all solutions. 7sin 3t Find all solutions on the interval [0, 2 ) sin t 15cos t sin.
7.1 Solving Trigonometric Equations with Identities In this section, we explore the techniques needed to solve more complex trig equations: By Factoring Using the Quadratic Formula Utilizing Trig Identities
More informationMATHEMATICS LEARNING AREA. Methods Units 1 and 2 Course Outline. Week Content Sadler Reference Trigonometry
MATHEMATICS LEARNING AREA Methods Units 1 and 2 Course Outline Text: Sadler Methods and 2 Week Content Sadler Reference Trigonometry Cosine and Sine rules Week 1 Trigonometry Week 2 Radian Measure Radian
More informationNo Calc. 1 p (seen anywhere) 1 p. 1 p. No Calc. (b) Find an expression for cos 140. (c) Find an expression for tan (a) (i) sin 140 = p A1 N1
IBSL /4 IB REVIEW: Trig KEY (0points). Let p = sin 40 and q = cos 0. Give your answers to the following in terms of p and/or q. (a) Write down an expression for (i) sin 40; (ii) cos 70. (b) Find an expression
More information6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities
Chapter 6: Trigonometric Identities 1 Chapter 6 Complete the following table: 6.1 Reciprocal, Quotient, and Pythagorean Identities Pages 290 298 6.3 Proving Identities Pages 309 315 Measure of
More informationADDITONAL MATHEMATICS
ADDITONAL MATHEMATICS 00 0 CLASSIFIED TRIGONOMETRY Compiled & Edited B Dr. Eltaeb Abdul Rhman www.drtaeb.tk First Edition 0 5 Show that cosθ + + cosθ = cosec θ. [3] 0606//M/J/ 5 (i) 6 5 4 3 0 3 4 45 90
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 information*n23494b0220* C3 past-paper questions on trigonometry. 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (2)
C3 past-paper questions on trigonometry physicsandmathstutor.com June 005 1. (a) Given that sin θ + cos θ 1, show that 1 + tan θ sec θ. (b) Solve, for 0 θ < 360, the equation tan θ + secθ = 1, giving your
More informationChapter 5 Analytic Trigonometry
Chapter 5 Analytic Trigonometry Overview: 5.1 Using Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Solving Trig Equations 5.4 Sum and Difference Formulas 5.5 Multiple-Angle and Product-to-sum
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 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 informationCHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS
CHINO VALLEY UNIFIED SCHOOL DISTRICT INSTRUCTIONAL GUIDE TRIGONOMETRY / PRE-CALCULUS Course Number 5121 Department Mathematics Qualification Guidelines Successful completion of both semesters of Algebra
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 informationCourse outline Mathematics: Methods ATAR Year 11
Course outline Mathematics: Methods ATAR Year 11 Unit 1 Sequential In Unit 1 students will be provided with opportunities to: underst the concepts techniques in algebra, functions, graphs, trigonometric
More information4 The Trigonometric Functions
Mathematics Learning Centre, University of Sydney 8 The Trigonometric Functions The definitions in the previous section apply to between 0 and, since the angles in a right angle triangle can never be greater
More informationName Date Period. Calculater Permitted MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
PreAP Precalculus Spring Final Exam Review Name Date Period Calculater Permitted MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplify the expression.
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 informationUnit 3 Trigonometry. 3.4 Graph and analyze the trigonometric functions sine, cosine, and tangent to solve problems.
1 General Outcome: Develop trigonometric reasoning. Specific Outcomes: Unit 3 Trigonometry 3.1 Demonstrate an understanding of angles in standard position, expressed in degrees and radians. 3.2 Develop
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 informationAlgebra II Standard Term 4 Review packet Test will be 60 Minutes 50 Questions
Algebra II Standard Term Review packet 2017 NAME Test will be 0 Minutes 0 Questions DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document.
More information6.1 The Inverse Sine, Cosine, and Tangent Functions Objectives
Objectives 1. Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function. 2. Find an Approximate Value of an Inverse Sine Function. 3. Use Properties of Inverse Functions to Find Exact Values
More informationAnalytic Trigonometry. Copyright Cengage Learning. All rights reserved.
Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.4 Basic Trigonometric Equations Copyright Cengage Learning. All rights reserved. Objectives Basic Trigonometric Equations Solving
More informationPhysicsAndMathsTutor.com
PhysicsAndMathsTutor.com physicsandmathstutor.com June 2005 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (b) Solve, for 0 θ < 360, the equation 2 tan 2 θ + secθ = 1, giving your
More information6.1 Solutions to Exercises
Last edited 3/1/13 6.1 Solutions to Exercises 1. There is a vertical stretch with a factor of 3, and a horizontal reflection. 3. There is a vertical stretch with a factor of. 5. Period:. Amplitude: 3.
More informationTopic Outline for Algebra 2 & and Trigonometry One Year Program
Topic Outline for Algebra 2 & and Trigonometry One Year Program Algebra 2 & and Trigonometry - N - Semester 1 1. Rational Expressions 17 Days A. Factoring A2.A.7 B. Rationals A2.N.3 A2.A.17 A2.A.16 A2.A.23
More informationSection 6.2 Notes Page Trigonometric Functions; Unit Circle Approach
Section Notes Page Trigonometric Functions; Unit Circle Approach A unit circle is a circle centered at the origin with a radius of Its equation is x y = as shown in the drawing below Here the letter t
More informationPhysicsAndMathsTutor.com
1. The diagram above shows the sector OA of a circle with centre O, radius 9 cm and angle 0.7 radians. Find the length of the arc A. Find the area of the sector OA. The line AC shown in the diagram above
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. and θ is in quadrant IV. 1)
Chapter 5-6 Review Math 116 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the fundamental identities to find the value of the trigonometric
More informationAlgebra II B Review 5
Algebra II B Review 5 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the measure of the angle below. y x 40 ο a. 135º b. 50º c. 310º d. 270º Sketch
More informationTrigonometric Functions. Section 1.6
Trigonometric Functions Section 1.6 Quick Review Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Radian
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 informationand sinθ = cosb =, and we know a and b are acute angles, find cos( a+ b) Trigonometry Topics Accuplacer Review revised July 2016 sin.
Trigonometry Topics Accuplacer Revie revised July 0 You ill not be alloed to use a calculator on the Accuplacer Trigonometry test For more information, see the JCCC Testing Services ebsite at http://jcccedu/testing/
More informationGrade 11 or 12 Pre-Calculus
Grade 11 or 12 Pre-Calculus Strands 1. Polynomial, Rational, and Radical Relationships 2. Trigonometric Functions 3. Modeling with Functions Strand 1: Polynomial, Rational, and Radical Relationships Standard
More informationAS Mathematics Assignment 9 Due Date: Friday 22 nd March 2013
AS Mathematics Assignment 9 Due Date: Friday 22 nd March 2013 NAME GROUP: MECHANICS/STATS Instructions to Students All questions must be attempted. You should present your solutions on file paper and submit
More informationSummer Review Packet for Students Entering AP Calculus BC. Complex Fractions
Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common
More informationFunctions and their Graphs
Chapter One Due Monday, December 12 Functions and their Graphs Functions Domain and Range Composition and Inverses Calculator Input and Output Transformations Quadratics Functions A function yields a specific
More informationCore 3 (A2) Practice Examination Questions
Core 3 (A) Practice Examination Questions Trigonometry Mr A Slack Trigonometric Identities and Equations I know what secant; cosecant and cotangent graphs look like and can identify appropriate restricted
More informationSummer 2017 Review For Students Entering AP Calculus AB/BC
Summer 2017 Review For Students Entering AP Calculus AB/BC Holy Name High School AP Calculus Summer Homework 1 A.M.D.G. AP Calculus AB Summer Review Packet Holy Name High School Welcome to AP Calculus
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 informationSection 8.5. z(t) = be ix(t). (8.5.1) Figure A pendulum. ż = ibẋe ix (8.5.2) (8.5.3) = ( bẋ 2 cos(x) bẍ sin(x)) + i( bẋ 2 sin(x) + bẍ cos(x)).
Difference Equations to Differential Equations Section 8.5 Applications: Pendulums Mass-Spring Systems In this section we will investigate two applications of our work in Section 8.4. First, we will consider
More informationGrade Math (HL) Curriculum
Grade 11-12 Math (HL) Curriculum Unit of Study (Core Topic 1 of 7): Algebra Sequences and Series Exponents and Logarithms Counting Principles Binomial Theorem Mathematical Induction Complex Numbers Uses
More informationLesson 33 - Trigonometric Identities. Pre-Calculus
Lesson 33 - Trigonometric Identities Pre-Calculus 1 (A) Review of Equations An equation is an algebraic statement that is true for only several values of the variable The linear equation 5 = 2x 3 is only
More information( and 1 degree (1 ) , there are. radians in a full circle. As the circumference of a circle is. radians. Therefore, 1 radian.
Angles are usually measured in radians ( c ). The radian is defined as the angle that results when the length of the arc of a circle is equal to the radius of that circle. As the circumference of a circle
More information~UTS. University of Technology, Sydney TO BE RETURNED AT THE END OF THE EXAMINATION. THIS PAPER MUST NOT BE REMOVED FROM THE EXAM CENTRE.
~UTS Cover Type B University of Technology, Sydney TO BE RETURNED AT THE END OF THE EXAMINATION. THIS PAPER MUST NOT BE REMOVED FROM THE EXAM CENTRE. SURNAME: FIRST NAME: STUDENTNUMBER: COURSE: AUTUMN
More informationSection 6.2 Trigonometric Functions: Unit Circle Approach
Section. Trigonometric Functions: Unit Circle Approach The unit circle is a circle of radius centered at the origin. If we have an angle in standard position superimposed on the unit circle, the terminal
More informationALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340
ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:
More informationFunctions Modeling Change A Preparation for Calculus Third Edition
Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1 CHAPTER
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 informationGiven an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
More informationUnit 6 Trigonometric Identities
Unit 6 Trigonometric Identities Prove trigonometric identities Solve trigonometric equations Prove trigonometric identities, using: Reciprocal identities Quotient identities Pythagorean identities Sum
More informationTRIGONOMETRIC FUNCTIONS. Copyright Cengage Learning. All rights reserved.
12 TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. 12.2 The Trigonometric Functions Copyright Cengage Learning. All rights reserved. The Trigonometric Functions and Their Graphs
More informationGiven an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
More informationMth 133 Trigonometry Review Problems for the Final Examination
Mth 1 Trigonometry Review Problems for the Final Examination Thomas W. Judson Stephen F. Austin State University Fall 017 Final Exam Details The final exam for MTH 1 will is comprehensive and will cover
More informationSCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self Study Course
SHOOL OF MTHEMTIS MTHEMTIS FOR PRT I ENGINEERING Self Study ourse MODULE TRIGONOMETRY Module Topics 1 The measurement of angles and conversion etween degrees and radians The elementary trigonometric ratios
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 informationMath 175: Chapter 6 Review: Trigonometric Functions
Math 175: Chapter 6 Review: Trigonometric Functions In order to prepare for a test on Chapter 6, you need to understand and be able to work problems involving the following topics. A. Can you sketch an
More informationSolutionbank C1 Edexcel Modular Mathematics for AS and A-Level
Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Find the values of x for which f ( x ) = x x is a decreasing function. f ( x ) = x x f ( x ) = x x Find f ( x ) and put
More information4-3 Trigonometric Functions on the Unit Circle
Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on
More informationMPE Review Section II: Trigonometry
MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the
More information