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NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 12 PRECALCULUS AND ADVANCED TOPICS Lesson 12 Inverse Trigonometric Functions S83 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 12 Inverse Trigonometric Functions Classwork Opening Exercise Use the graphs of the sine cosine and tangent functions to answer each of the following questions a State the domain of each function b Would the inverse of the sine cosine or tangent functions also be functions Explain c For each function select a suitable domain that will make the function invertible NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 12 PRECALCULUS AND ADVANCED TOPICS Lesson 12 Inverse Trigonometric Functions S84 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Example 1 Consider the function 𝑓𝑥 sin𝑥 𝜋 2 𝑥 𝜋 2 a State the domain and range of this function b Find the equation of the inverse function c State the domain and range of the inverse Exercises 13 1 Write an equation for the inverse cosine function and state its domain and range 2 Write an equation for the inverse tangent function and state its domain and range NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 12 PRECALCULUS AND ADVANCED TOPICS Lesson 12 Inverse Trigonometric Functions S85 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Evaluate each of the following expressions without using a calculator Use radian measures a sin1 3 2 b sin1 3 2 c cos1 3 2 d cos1 3 2 e sin11 f sin11 g cos11 h cos11 i tan11 j tan11 Example 2 Solve each trigonometric equation such that 0 𝑥 2𝜋 Round to three decimal places when necessary a 2cos𝑥 1 0 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 12 PRECALCULUS AND ADVANCED TOPICS Lesson 12 Inverse Trigonometric Functions S86 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License b 3 sin𝑥 2 0 Exercises 48 4 Solve each trigonometric equation such that 0 𝑥 2𝜋 Give answers in exact form a 2cos𝑥 1 0 b tan𝑥 3 0 c sin2𝑥 1 0 5 Solve each trigonometric equation such that 0 𝑥 2𝜋 Round answers to three decimal places a 5 cos𝑥 3 0 b 3 cos𝑥 5 0 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 12 PRECALCULUS AND ADVANCED TOPICS Lesson 12 Inverse Trigonometric Functions S87 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License c 3 sin𝑥 1 0 d tan𝑥 0115 6 A particle is moving along a straight line for 0 𝑡 18 The velocity of the particle at time 𝑡 in seconds is given by the function 𝑣𝑡 cos 𝜋 5 𝑡 Find the times on the interval 0 𝑡 18 where the particle is at rest 𝑣𝑡 0 7 In an amusement park there is a small Ferris wheel called a kiddie wheel for toddlers The formula 𝐻𝑡 10 sin 2𝜋 𝑡 1 4 15 models the height 𝐻 in feet of the bottommost car 𝑡 minutes after the wheel begins to rotate Once the ride starts it lasts 4 minutes a What is the initial height of the car b How long does it take for the wheel to make one full rotation c What is the maximum height of the car NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 12 PRECALCULUS AND ADVANCED TOPICS Lesson 12 Inverse Trigonometric Functions S88 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License d Find the times on the interval 0 𝑡 4 when the car is at its maximum height 8 Many animal populations fluctuate periodically Suppose that a wolf population over an 8year period is given by the function 𝑊𝑡 800sin 𝜋 4 𝑡 2200 where 𝑡 represents the number of years since the initial population counts were made a Find the times on the interval 0 𝑡 8 such that the wolf population equals 2500 b On what time interval during the 8year period is the population below 2000 c Why would an animal population be an example of a periodic phenomenon NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 12 PRECALCULUS AND ADVANCED TOPICS Lesson 12 Inverse Trigonometric Functions S89 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Solve the following equations Approximate values of the inverse trigonometric functions to the thousandths place where 𝑥 refers to an angle measured in radians a 5 6 cos𝑥 b 1 2 2 cos 𝑥 𝜋 4 1 c 1 cos3𝑥 1 d 12 05 cos𝜋𝑥 09 e 7 9 cos𝑥 4 f 2 3 sin𝑥 g 1 sin 𝜋𝑥1 4 1 h 𝜋 3 sin5𝑥 2 2 i 1 9 sin𝑥 4 j cos𝑥 sin𝑥 k sin1cos𝑥 𝜋 3 l tan𝑥 3 m 1 2 tan5𝑥 2 3 n 5 15 tan𝑥 3 2 Fill out the following tables 𝒙 𝐬𝐢𝐧𝟏𝒙 𝐜𝐨𝐬𝟏𝒙 𝒙 𝐬𝐢𝐧𝟏𝒙 𝐜𝐨𝐬𝟏𝒙 1 0 3 2 1 2 2 2 2 2 1 2 3 2 1 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 12 PRECALCULUS AND ADVANCED TOPICS Lesson 12 Inverse Trigonometric Functions S90 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 3 Let the velocity 𝑣 in miles per second of a particle in a particle accelerator after 𝑡 seconds be modeled by the function 𝑣 tan 𝜋𝑡 6000 𝜋 2 on an unknown domain a What is the 𝑡value of the first vertical asymptote to the right of the 𝑦axis b If the particle accelerates to 99 of the speed of light before stopping then what is the domain Note 𝑐 186000 Round your solution to the tenthousandths place c How close does the domain get to the vertical asymptote of the function d How long does it take for the particle to reach the velocity of Earth around the sun about 185 miles per second e What does it imply that 𝑣 is negative up until 𝑡 3000 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 1 PRECALCULUS AND ADVANCED TOPICS Lesson 1 Special Triangles and the Unit Circle S1 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 1 Special Triangles and the Unit Circle Classwork Example 1 Find the following values for the rotation 𝜃 𝜋 3 around the carousel Create a sketch of the situation to help you Interpret what each value means in terms of the position of the rider a sin𝜃 b cos𝜃 c tan𝜃 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 1 PRECALCULUS AND ADVANCED TOPICS Lesson 1 Special Triangles and the Unit Circle S2 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exercise 1 Assume that the carousel is being safety tested and a safety mannequin is the rider The ride is being stopped at different rotation values so technicians can check the carousels parts Find the sine cosine and tangent for each rotation indicated and explain how these values relate to the position of the mannequin when the carousel stops at these rotation values Use your carousel models to help you determine the values and sketch your model in the space provided a 𝜃 𝜋 4 b 𝜃 𝜋 6 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 1 PRECALCULUS AND ADVANCED TOPICS Lesson 1 Special Triangles and the Unit Circle S3 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Example 2 Use your understanding of the unit circle and trigonometric functions to find the values requested a sin 𝜋 3 b tan 5𝜋 4 Exercise 2 Use your understanding of the unit circle to determine the values of the functions shown a sin 11𝜋 6 b cos 3𝜋 4 c tan𝜋 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 1 PRECALCULUS AND ADVANCED TOPICS Lesson 1 Special Triangles and the Unit Circle S4 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Complete the chart below 𝜽 𝝅 𝟔 𝝅 𝟒 𝝅 𝟑 𝐬𝐢𝐧𝜽 𝐜𝐨𝐬𝜽 𝐭𝐚𝐧𝜽 2 Evaluate the following trigonometric expressions and explain how you used the unit circle to determine your answer a cos 𝜋 𝜋 3 b sin 𝜋 𝜋 4 c sin 2𝜋 𝜋 6 d cos 𝜋 𝜋 6 e cos 𝜋 𝜋 4 f cos 2𝜋 𝜋 3 g tan 𝜋 𝜋 4 h tan 𝜋 𝜋 6 i tan 2𝜋 𝜋 3 3 Rewrite the following trigonometric expressions in an equivalent form using 𝜋 𝜃 𝜋 𝜃 or 2𝜋 𝜃 and evaluate a cos 𝜋 3 b cos 𝜋 4 c sin 𝜋 6 d sin 4𝜋 3 e tan 𝜋 6 f tan 5𝜋 6 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 1 PRECALCULUS AND ADVANCED TOPICS Lesson 1 Special Triangles and the Unit Circle S5 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 4 Identify the quadrant of the plane that contains the terminal ray of a rotation by 𝜃 if 𝜃 satisfies the given conditions a sin𝜃 0 and cos𝜃 0 b sin𝜃 0 and cos𝜃 0 c sin𝜃 0 and tan𝜃 0 d tan𝜃 0 and sin𝜃 0 e tan𝜃 0 and sin𝜃 0 f tan𝜃 0 and cos𝜃 0 g cos𝜃 0 and tan𝜃 0 h sin𝜃 0 and cos𝜃 0 5 Explain why sin2𝜃 cos2𝜃 1 6 Explain how it is possible to have sin𝜃 0 cos𝜃 0 and tan𝜃 0 For which values of 𝜃 between 0 and 2𝜋 does this happen 7 Duncan says that for any real number 𝜃 tan𝜃 tan𝜋 𝜃 Is he correct Explain how you know 8 Given the following trigonometric functions identify the quadrant in which the terminal ray of 𝜃 lies in the unit circle shown below Find the other two trigonometric functions of 𝜃 of sin𝜃 cos𝜃 and tan𝜃 a sin𝜃 1 2 and cos𝜃 0 b cos𝜃 1 2 and sin𝜃 0 c tan𝜃 1 and cos𝜃 0 d sin𝜃 3 2 and cot𝜃 0 e tan𝜃 3 and cos𝜃 0 f sec𝜃 2 and sin𝜃 0 g cot𝜃 3 and csc𝜃 0 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 1 PRECALCULUS AND ADVANCED TOPICS Lesson 1 Special Triangles and the Unit Circle S6 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 9 Toby thinks the following trigonometric equations are true Use 𝜃 𝜋 6 𝜋 4 and 𝜋 3 to develop a conjecture whether or not he is correct in each case below a sin𝜃 cos 𝜋 2 𝜃 b cos𝜃 sin 𝜋 2 𝜃 10 Toby also thinks the following trigonometric equations are true Is he correct Justify your answer a sin 𝜋 𝜋 3 sin𝜋 sin 𝜋 3 b cos 𝜋 𝜋 3 cos𝜋 cos 𝜋 3 c tan 𝜋 3 𝜋 6 tan 𝜋 3 tan 𝜋 6 d sin 𝜋 𝜋 6 sin𝜋 sin 𝜋 6 e cos 𝜋 𝜋 4 cos𝜋 cos 𝜋 4 NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 PRECALCULUS AND ADVANCED TOPICS Lesson 21 Vectors and the Equation of a Line S156 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 21 Vectors and the Equation of a Line Classwork Opening Exercise a Find three different ways to write the equation that represents the line in the plane that passes through points 12 and 2 1 b Graph the line through point 11 with slope 2 Exercises 1 Consider the line ℓ in the plane given by the equation 3𝑥 2𝑦 6 a Sketch a graph of line ℓ on the axes provided b Find a point on line ℓ and the slope of line ℓ NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 PRECALCULUS AND ADVANCED TOPICS Lesson 21 Vectors and the Equation of a Line S157 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License c Write a vector equation for line ℓ using the information you found in part b d Write parametric equations for line ℓ e Verify algebraically that your parametric equations produce points on line ℓ 2 Olivia wrote parametric equations 𝑥𝑡 4 2𝑡 and 𝑦𝑡 3 3𝑡 Are her equations correct What did she do differently from you 3 Convert the parametric equations 𝑥𝑡 2 3𝑡 and 𝑦𝑡 4 𝑡 into slopeintercept form NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 PRECALCULUS AND ADVANCED TOPICS Lesson 21 Vectors and the Equation of a Line S158 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 4 Find parametric equations to represent the line that passes through point 429 and has direction vector 𝐯 2 1 3 5 Find a vector form of the equation of the line given by the parametric equations 𝑥𝑡 3𝑡 𝑦𝑡 4 2𝑡 𝑧𝑡 3 𝑡 NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 PRECALCULUS AND ADVANCED TOPICS Lesson 21 Vectors and the Equation of a Line S159 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson Summary Lines in the plane and lines in space can be described by either a vector equation or a set of parametric equations Let ℓ be a line in the plane that contains point 𝑥1 𝑦1 and has direction vector 𝐯 𝑎 𝑏 If the slope of line ℓ is defined then 𝑚 𝑏 𝑎 A vector form of the equation that represents line ℓ is Parametric equations that represent line ℓ are Let ℓ be a line in space that contains point 𝑥1 𝑦1 𝑧1 and has direction vector 𝐯 𝑎 𝑏 𝑐 A vector form of the equation that represents line ℓ is Parametric equations that represent line ℓ are 𝑥 𝑦 𝑥1 𝑦1 𝑎 𝑏 𝑡 𝑥𝑡 𝑥1 𝑎𝑡 𝑦𝑡 𝑦1 𝑏𝑡 𝑥 𝑦 𝑧 𝑥1 𝑦1 𝑧1 𝑎 𝑏 𝑐 𝑡 𝑥𝑡 𝑥1 𝑎𝑡 𝑦𝑡 𝑦1 𝑏𝑡 𝑧𝑡 𝑧1 𝑐𝑡 Problem Set 1 Find three points on the line in the plane with parametric equations 𝑥𝑡 4 3𝑡 and 𝑦𝑡 1 1 3 𝑡 2 Find vector and parametric equations to represent the line in the plane with the given equation a 𝑦 3𝑥 4 b 2𝑥 5𝑦 10 c 𝑦 𝑥 d 𝑦 2 3𝑥 1 3 Find vector and parametric equations to represent the following lines in the plane a the 𝑥axis b the 𝑦axis NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 PRECALCULUS AND ADVANCED TOPICS Lesson 21 Vectors and the Equation of a Line S160 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License c the horizontal line with equation 𝑦 4 d the vertical line with equation 𝑥 2 e the horizontal line with equation 𝑦 𝑘 for a real number 𝑘 f the vertical line with equation 𝑥 ℎ for a real number ℎ 4 Find the pointslope form of the line in the plane with the given parametric equations a 𝑥𝑡 2 4𝑡 𝑦𝑡 3 7𝑡 b 𝑥𝑡 2 2 3 𝑡 𝑦𝑡 6 6𝑡 c 𝑥𝑡 3 𝑡 𝑦𝑡 3 d 𝑥𝑡 𝑡 𝑦𝑡 𝑡 5 Find vector and parametric equations for the line in the plane through point 𝑃 in the direction of vector 𝐯 a 𝑃 15 𝐯 2 1 b 𝑃 00 𝐯 4 4 c 𝑃 3 1 𝐯 1 2 6 Determine if the point 𝐴 is on the line ℓ represented by the given parametric equations a 𝐴 31 𝑥𝑡 1 2𝑡 and 𝑦𝑡 3 2𝑡 b 𝐴 00 𝑥𝑡 3 6𝑡 and 𝑦𝑡 2 4𝑡 c 𝐴 23 𝑥𝑡 4 2𝑡 and 𝑦𝑡 4 𝑡 d 𝐴 25 𝑥𝑡 12 2𝑡 and 𝑦𝑡 15 2𝑡 7 Find three points on the line in space with parametric equations 𝑥𝑡 4 2𝑡 𝑦𝑡 6 𝑡 and 𝑧𝑡 𝑡 8 Find vector and parametric equations to represent the following lines in space a the 𝑥axis b the 𝑦axis c the 𝑧axis NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 21 PRECALCULUS AND ADVANCED TOPICS Lesson 21 Vectors and the Equation of a Line S161 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM2TE130082015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 9 Convert the equation given in vector form to a set of parametric equations for the line ℓ a 𝑥 𝑦 𝑧 1 1 1 2 3 4 𝑡 b 𝑥 𝑦 𝑧 3 0 0 0 1 2 𝑡 c 𝑥 𝑦 𝑧 5 0 2 4 3 8 𝑡 10 Find vector and parametric equations for the line in space through point 𝑃 in the direction of vector 𝐯 a 𝑃 143 𝐯 3 6 2 b 𝑃 222 𝐯 1 1 1 c 𝑃 000 𝐯 4 4 2 11 Determine if the point 𝐴 is on the line ℓ represented by the given parametric equations a 𝐴 311 𝑥𝑡 5 𝑡 𝑦𝑡 5 3𝑡 and 𝑧𝑡 9 4𝑡 b 𝐴 102 𝑥𝑡 7 2𝑡 𝑦𝑡 3 𝑡 and 𝑧𝑡 4 𝑡 c 𝐴 532 𝑥𝑡 8 𝑡 𝑦𝑡 𝑡 and 𝑧𝑡 4 2𝑡 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Revisiting the Graphs of the Trigonometric Functions S75 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Lesson 11 Revisiting the Graphs of the Trigonometric Functions Classwork Opening Exercise Graph each of the following functions on the interval 2𝜋 𝑥 4𝜋 by making a table of values The graph should show all key features intercepts asymptotes relative maxima and minima a 𝑓𝑥 sin𝑥 𝒙 𝐬𝐢𝐧𝒙 b 𝑓𝑥 cos𝑥 𝒙 𝐜𝐨𝐬𝒙 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Revisiting the Graphs of the Trigonometric Functions S76 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Exploratory ChallengeExercises 17 1 Consider the trigonometric function 𝑓𝑥 tan𝑥 a Rewrite tan𝑥 as a quotient of trigonometric functions Then state the domain of the tangent function b Why is this the domain of the function c Complete the table 𝒙 2𝜋 3𝜋 2 𝜋 𝜋 2 0 𝜋 2 𝜋 3𝜋 2 2𝜋 5𝜋 2 3𝜋 7𝜋 2 4𝜋 𝐭𝐚𝐧𝒙 d What will happen on the graph of 𝑓𝑥 tan𝑥 at the values of 𝑥 for which the tangent function is undefined e Expand the table to include angles that have a reference angle of 𝜋 4 𝒙 7𝜋 2 5𝜋 2 3𝜋 2 𝜋 2 𝜋 2 3𝜋 2 5𝜋 2 7𝜋 2 9𝜋 2 11𝜋 2 13𝜋 2 15𝜋 2 𝐭𝐚𝐧𝒙 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Revisiting the Graphs of the Trigonometric Functions S77 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License f Sketch the graph of 𝑓𝑥 tan𝑥 on the interval 2𝜋 𝑥 4𝜋 Verify by using a graphing utility 2 Use the graphs of the sine cosine and tangent functions to answer each of the following a How do the graphs of the sine and cosine functions support the following identities for all real numbers 𝑥 sin𝑥 sin𝑥 cos𝑥 cos𝑥 b Use the symmetry of the graph of the tangent function to write an identity Explain your answer c How do the graphs of the sine and cosine functions support the following identities for all real numbers 𝑥 sin𝑥 2𝜋 sin𝑥 cos𝑥 2𝜋 cos𝑥 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Revisiting the Graphs of the Trigonometric Functions S78 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License d Use the periodicity of the tangent function to write an identity Explain your answer 3 Consider the function 𝑓𝑥 cos 𝑥 𝜋 2 a Graph 𝑦 𝑓𝑥 by using transformations of functions b Based on your graph write an identity 4 Verify the identity sin 𝑥 𝜋 2 cos𝑥 for all real numbers 𝑥 by using a graph NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Revisiting the Graphs of the Trigonometric Functions S79 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 5 Use a graphing utility to explore the graphs of the family of functions in the form 𝑓𝑥 𝐴sin𝜔𝑥 ℎ 𝑘 Write a summary of the effect that changing each parameter has on the graph of the sine function a 𝐴 b 𝜔 c ℎ d 𝑘 6 Graph at least one full period of the function 𝑓𝑥 3sin 1 3 𝑥 𝜋 2 Label the amplitude period and midline on the graph NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Revisiting the Graphs of the Trigonometric Functions S80 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 7 The graph and table below show the average monthly high and low temperature for Denver Colorado source httpwwwrssweathercomclimateColoradoDenver a Why would a sinusoidal function be appropriate to model this data b Write a function to model the average monthly high temperature as a function of the month c What does the midline represent within the context of the problem d What does the amplitude represent within the context of the problem e Name a city whose temperature graphs would have a smaller amplitude Explain your reasoning f Name a city whose temperature graphs would have a larger vertical shift Explain your reasoning NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Revisiting the Graphs of the Trigonometric Functions S81 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License Problem Set 1 Sketch the graph of 𝑦 sin𝑥 on the same set of axes as the function 𝑓𝑥 sin4𝑥 Explain the similarities and differences between the two graphs 2 Sketch the graph of 𝑦 sin 𝑥 2 on the same set of axes as the function 𝑔𝑥 3sin 𝑥 2 Explain the similarities and differences between the two graphs 3 Indicate the amplitude frequency period phase shift horizontal and vertical translations and equation of the midline Graph the function on the same axes as the graph of the cosine function 𝑓𝑥 cos𝑥 Graph at least one full period of each function 𝑔𝑥 cos 𝑥 3𝜋 4 4 Sketch the graph of the pairs of functions on the same set of axes 𝑓𝑥 sin4𝑥 𝑔𝑥 sin4𝑥 2 5 The graph and table below show the average monthly high and low temperature for Denver Colorado source httpwwwrssweathercomclimateColoradoDenver Write a function to model the average monthly low temperature as a function of the month Extension 6 Consider the cosecant function a Use technology to help you sketch 𝑦 csc𝑥 for 0 𝑥 4𝜋 4 𝑦 4 b What do you notice about the graph of the function Compare this to your knowledge of the graph of 𝑦 sin𝑥 NYS COMMON CORE MATHEMATICS CURRICULUM M4 Lesson 11 PRECALCULUS AND ADVANCED TOPICS Lesson 11 Revisiting the Graphs of the Trigonometric Functions S82 This work is derived from Eureka Math and licensed by Great Minds 2015 Great Minds eurekamathorg This file derived from PreCalM4TE130102015 This work is licensed under a Creative Commons AttributionNonCommercialShareAlike 30 Unported License 7 Consider the secant function a Use technology to help you sketch 𝑦 sec𝑥 for 0 𝑥 4𝜋 4 𝑦 4 b What do you notice about the graph of the function Compare this to your knowledge of the graph of 𝑦 cos𝑥 8 Consider the cotangent function a Use technology to help you sketch 𝑦 cot𝑥 for 0 𝑥 2𝜋 4 𝑦 4 b What do you notice about the graph of the function Compare this to your knowledge of the graph of 𝑦 tan𝑥 the rider is approximately units in front of the carousels center when it stops the rider is approximately units to the right of the carousels center when it stops the frontback distance of the rider is equal to its rightleft distance when it stops the rider is approximately unit in front of the carousels center when it stops the rider is approximately units to the right of the carousels center when it stops the frontback to rightleft ratio of the rider is when it stops a The slope of lines for the graphs shown are 1 X 0 X 0 2 The limits taken are oo to oo 3 The domain of cos x is symmetric with respect the y axis since cos x cos x The range of cosine is symmetric with respect to y axis since y cos x is symmetric and cos x cos x The period of cosine is symmetric with respect to x axis since cos x 2 pi cos x 4 The function f x sin x has the y axis as an axis of symmetry since sin x sin x 5 The function f x sin x has the point 0 0 as a point of symmetry Since sin x sin x 6 a The graph of sin x is symmetric with respect the origin and sine is not one to one function b Graph of 2 pi 3 sin x 2 pi 3 y 2 pi 3 4 pi 3 c Properties similar to 3 above where cos function inverse is an onto function while sine is onto but the range are different for symmetric with respect of origin 7 Graphs of sine cos based on the formulas function has point at x and also at t are a graph of cos 3 x b Cos function is periodic c Function sin x d function y sin 4 x e cos 3x The special function has four functions 1 periodic sounds of the wind 2 Appearance of sounds in ear 3 Resonation sounds of the sound 4 Vibration sounds of the wind of the wind of the air 5 Vibration sounds of wind 6 Pitch and power of noise d Vertical axis of noise is frequency Hz 3 amplitude 7 db power 8 a The data about noisy plants b The 24 h cycle 3 has 1 Resonator 2 Nodes belt is danger c The noises depend on plant and other parameters in degree temperature or wind d The natural resonator used by the plant under this condition is the one having resonance Hence plant will use this resonance with degree of noise longer e Some types of resonation is transducers used with a bearing between holding the resonator name of some of those have recording module included the mic f Human it would nice a resonator even when outside breathing sounds disease this is place letting the air and material and trachea 13 2 y cos1 x D 1 x 1 R 0 y Π 3 y tan1 x D R R Π2 y Π2 3 a Π3 b Π3 c Π6 d 5Π6 e Π2 f Π2 g 0 h Π j Π4 J Π4 4 a cos x 12 x 3Π4 5Π4 b tan x 3 x Π3 4Π3 c sin x 1 x Π2 3Π2 5 a cos x 35 x cos1 35 x 0927 5356 b No real solution c sin x 13 x sin1 13 x 03401 2802 d tan x 0115 x tan1 0115 x 3027 6169 6 The particle is at t 25 s t 75 s t 125 s and t 175 s a a 5 ft b 1 minute c 25 ft d sin 2 π t 14 1 t 05 15 25 and 35 minutes 8 a t 0489 3511 The wolf population equals 2500 after approximately 05 years and again after 35 years b W t 2000 at t 4332 and t 7678 The wolf population is below 2000 on the interval 4332 7678 c An animal population might increase while their food source is plentiful Then when the population is too large there is less food and the population begins to decrease The cycle repeats 1 a 3x 2y 6 points 2 0 and 0 3 vector v 1 1 vector v 3 2 b x 0 2y 6 y 3 P 0 3 The slope m 32 c x t y 6 3t2 3 15t x y 0 3 1 15 t d Equal to c xt t yt 3 15t e 3t 23 15t 3t 6 3t 6 OK x t and y 3 15 t is in the line C 2 xt 4t 2t yt 3 3t 24 2t 23 3t 12 6t 6 6t 6 OK She used the point 4 3 and the vector 2 3 3 The line pass through 2 4 with slope m 13 Thus y 4 x 2 13 3y 12 x 2 3y x 14 y 13 x 143 4 x y z 4 2 9 2 1 3 t xt 4 2t yt 2 t zt 9 3t 5 x y z 0 4 3 3 2 1 t