( x ) ( a) 2. Chapter 2: Polynomial Functions. Essential Questions. What is a polynomial? Why do we study polynomials as a group?

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1 Chapter : Polynomial Functions Essential Questions What is a polynomial? Why do we study polynomials as a group? Why is it important to be able to find the roots/zeros/-intercepts of a polynomial? What is an imaginary number? Why do we study imaginary numbers? What does it mean for a function to have imaginary roots/zeros? Why is it important to be able to model a set of data using a function? Learning Targets Determine whether a function is a polynomial. Find the comple roots/zeros/-intercepts of a polynomial. Graph all polynomials. Write all polynomials as a product of their prime factors. Determine whether a function is rational. Determine what degree polynomial should be used to model different sets of data. Perform operations on comple numbers. Homework Section. Quadratic Functions Describe, in words, how the graph of each function is related to the graph of y =.. ( a) y = ( b) y = + = ( ) d y ( ) ( c) y. ( a) y = ( ) = + ( b) y = + = + d y ( ) ( c) y ( ) ( ) = + + Sketch the graph of the quadratic function. Identify the verte and -intercept(s).. f ( ) = 7. f ( ) = 6 ( ) 5. f ( ) = f ( ) = f ( ) = +

2 Write the equation in standard form of the quadratic function that has the indicated verte and travels through the given point. 8. Verte: (, -) and Point: (, ) Verte:, 0 and Point:, 0. A rancher has 00 feet of fencing. He plans on enclosing a rectangular area and then fencing it off into two congruent pens. What are the dimensions of the enclosed rectangular area that will produce the maimum area?. The path of a diver is given by y = + + where y is the height in feet and is the horizontal 9 9 distance in feet from the end of the diving board. What is the maimum height of the diver? Section. Polynomial Functions of Higher Degree For eercises -8, match the polynomial functions with its graph.. f ( ) = + (a) (b). ( ) = f. ( ) = 5 f. f ( ) = + 5. f ( ) = + (c) (d) 6. f ( ) = + 7. f ( ) = ( ) f = + (e) (f) (g) Graph the functions. Make sure to label their roots. 9. f ( ) = 0. f ( ) = ( 9) 5. ( ) = f. f ( ) = f ( ) =. f ( ) =

3 Find a polynomial function that has the given zeros. 5. 7, 6. 0,, 5 7.,, 0,, , 6 9. An open bo is to be made from a square piece of material 6 centimeters on a side by cutting equal squares with sides of length from the corners and turning up the sides. Find the formula for the volume of the bo and determine the dimensions of the bo that will maimize the volume. Section. Real Zeros of Polynomial Functions Use synthetic division to divide the polynomials.. ( ) ( + ). ( ) ( + ). ( ) ( + 6). ( + + ) ( + ) 5. ( 79) ( 9) Use synthetic division to find each function value. 6. f ( ) = + 0 ( a) f () ( b) f ( ) 6 7. f ( ) = + + ( a) f () ( ) ( ) ( c) f b f ( ) ( ) ( d) f (8) c f ( d) f ( ) Find all of the roots of the given polynomials. Write each function as the product of linear factors. 8. f ( ) = f ( ) = + 0 Find all of the solutions of the polynomial equations = f ( ) = f ( ) = = 0 Section. Comple Numbers Perform the given operation and write your answer in standard form.. ( i) ( 6 i) + +. ( 7 + 8) ( + i ). + ( i) + 0i ( i) ( ) i i ( 75 ) 8. (6 i)( i) 9. i(6 i) 0. ( + 5)( 7 0 )

4 . ( i) ( + i) 5. i 8 7i.. i i 5. 5i ( + i ) 7. i i 6. i i i ( ) 6 8. Section.5 The Fundamental Theorem of Algebra Find all zeros of the functions.. f ( ) = ( + ). f ( ) = ( + ) ( ). f ( ) = ( + 9)( + i)( i). f ( ) = ( )( )( i)( + i) Find all of the zeros of the functions. Then graph them and write the functions as the product of linear factors. 5. f ( ) = f ( ) = f ( ) = f ( ) = f ( ) = Find a polynomial with real coefficients that has the given zeros. 0., i, i., 6 + 5i. 0, 0,, + i Use the given root to find all of the roots of the function.. ( ) = , = f Root i. ( ) 0, f = + + Root = + i 5. ( ) = , = f Root i

5 Chapter Review For -, sketch the graph of the quadratic functions. Make sure to label the verte and any intercepts.. f ( ) = +. f ( ) = + ( ). f ( ) = +. f ( ) 5 Find f ( ). = + f = ( ) Find f Write the comple number in standard form. 5. ( ) i 8. + i 7. 5 i ( 8 i) ( i)( i) Find all of the zeros of the given polynomials. 0. f ( ) = f ( ) = Sketch the graph of the polynomial functions. Make sure to label any intercepts and their multiplicities. = ( + )( ). f ( ) = ( ) ( + ) ( ). f ( ) 5. f ( ) = ( ) f = + 6. Find the polynomial function with the given zeros, multiply and simplify your answer. a), -, -i b) -i, c) +i, 0, 0, d) --i,,, Write an equation that represents the graph below. Multiply and simplify your function. 7. f ( ) =

6 Match the polynomial function to the given zeros and multiplicities. a) b) c) d) 8. - (multiplicity of ), (multiplicity of ) 9. - (multiplicity of ), (multiplicity of ) 0. - (multiplicity of ), (multiplicity of ). - (multiplicity of ), (multiplicity of ). A function that is defined by the set of ordered pairs {(,), (,), (6,)} has domain {,, 6}. What is the domain of the function defined by the set of ordered pairs {(0,), (,), (, -)}? a. {} b. {-, } c. {-, 0, } d. {0,, } e. {-, 0,, }. Which of the following epressions is equivalent to + 6 5? a. ( +5)( -) b. ( -5)( +) c. ( +6-5) d. ( -5)( +) e. ( +5)( -). What is the sum of the solutions of the equation +-6=0? a. -6 b. - c. - d. 0 e.

7 Chapter 7: Solving Systems of Equations Essential Questions What is a system of equations? What does a solution to a system of equation represent in terms of its equations? Its graphs? Which method do you prefer when trying to solve a system of equations? Why would you choose one method over the other? Learning Targets Solve systems of equations using substitution, linear combination, and graphing. Homework Section 7. Solving Systems of Equations Solve the systems of equations using substitution. + y =. 5 y =.5 +.y = y =.6 y = 5. + y = 7. y = y = y = 9. y = 6 y = 0. + y = 0 + y = 0. y = + y = 6. y = 6 + y = 0 8. y = 0 0. A small fast-food restaurant invests $5000 to produce a new food item that will sell for $.9. Each item can be produced for $.6. How many items must be sold to break even? Section 7. Systems of Linear Equations in Two Variables Solve the systems of equations using elimination. + 5y = y = 0 + y =. 6 y = 0 + y =. 6 + y = y + + =. y = 5

8 .05.0 y = y =.6 y = 8 7. y = y = 6 6. y = + y = y = 9.Find a system of equations that has (, -) as a solution tickets were sold at the last 5FD show. The tickets for adults cost $5 and the tickets for students cost $. If the receipts for the show totaled $0, how many of each type of ticket was sold? Section 7. Multivariable Linear Systems Solve the systems of equations. + y + z =. + y = + y + z = 8 + y + z =. y + 6z = y + z = 6 + y + z = y + 8z = 6 + 8y + 8z = 5 + z = 7. y + z = 7 y 7z = 9 y 6z = 9. + y + 6z = y 5z = + y z =. y + z = 9 + y + z = 5 y + z =. + y z = 7 y + z = + y z = 6. + z = 0 + y z = 8 + y + z = 8. y 5z = 7 + y z = y + z = 0 + y z = 0. 0 y + z = y z = 0 An object moving vertically is at the given heights at the given times. Find the position equation s = at + v0t + s0 for the object.. At t = sec, s = 8 ft At t = sec, s = 6 ft At t = sec, s = 8 ft. At t = sec, s = ft At t = sec, s = 00 ft At t = sec, s = 6 ft

9 Chapter 7 Review Solve the systems of equations.. + y = y = + y = 0. + y = 7 + y =. y = 0 + 5y = 5. 9 y = 7. = y + y = 6 y = 9. y z = + y + z =. + y = y = y = 8 6. y = y + z = 5 8. y + z = 9 8y + 6z = + y z = 0 0. y + z = 6 + z = 7. Mr. Wu spent his three day weekend selling lemonade on the streets. He spent $0 on signs, a chair, and a table for his stand. If he sold each cup of lemonade for $.5, but it cost $.09 total for the cup, ice, lemons, and sugar, how many cups of lemonade would Mr. Wu have to sell to break even?. Mr. Coulson has $5000 to invest. He splits the money into an IRA and a 0b, which have a return of % and 6% respectively. If he earns $0 total after one year, how much did Mr. Coulson invest into each account?. Mrs. Hopkins, Mr. Coulson, and Mr. Wu went to si flags together this summer. They bought three tickets, si cheeseburgers, and two jumbo ice cream cones for a total of $0. One ticket cost as much as all of the cheeseburgers, and you could buy cheeseburgers for the cost of one jumbo ice cream cone. How much does a jumbo ice cream cone cost?. A plumber s total charge includes a fied service charge plus an hourly rate for the job. If the total charge is $0 for a -hour job and $00 for a 5-hour job, what is the total charge for an 8-hour job? a. $0 b. $90 c. $60 d. $50 e.$0 5. As a fund-raiser, a local youth group sold boes of regular popcorn for $5 each and boes of caramel popcorn for $8 each. Altogether, they sold 60 boes for $,00. How many boes of caramel popcorn did they sell? a. 0 b. c. 60 d. 80 e On a recent test, some questions were worth points each and the rest were worth points each. Tuan answered correctly the same number of -point questions as -point questions and earned a score of 60 points. How many -point questions did he answer correctly? a. 6 b. 0 c. 0 d. e Becky has 76 solid-colored disks that are either red, blue, or green. She lines them up on the floor and finds that there are more red disks than green and 6 more green disks than blue. How many red disks does she have? a. 0 b. 0 c. d. 6 e. 0

10 Chapter : Eponential & Logarithmic Functions Essential Questions What is an eponential function? What is eponentiating? What is a logarithmic function? How are eponential and logarithmic functions similar to other functions that we have studied? How are eponential and logarithmic functions different from other functions? Learning Targets Determine whether a function is eponential or logarithmic. Graph all eponential and logarithmic functions. Solve eponential and logarithmic equations. Determine whether a set of data can best be modeled using a linear, polynomial, rational, eponential, or logarithmic function. Homework Sections. - Eponential Functions & Graphs. Graph the function. Identify any asymptotes. (a) f ( ) = ( ) (b) f ( ) = ( ) (c) f ( ) = ( ). 5. Match the function to the graph. (a) (b) (c) (d) +. f ( ) =. f ( ) =. f ( ) = 5. f ( ) = Sketch the graph of the function. 6. f ( ) = + 7. g( ) = + e 8. s( t) = 0.t e 9. Compound Interest: Complete the table for balance A using the correct compound interest formula. P = $000 r = 6% t = 0 years n 65 Continuous A

11 0. Radioactive Decay: Let Q represent a mass of Carbon ( C), in grams, whose half-life is 570 years. The t quantity present after t years is given by 0( ) 570 Q =. (a) Determine the initial quantity (t = 0) (b) Determine the quantity present after 000 years. (c) Graph the function Q over the interval from t = 0 to t = 0,000 (d) When will this quantity of C be 0 grams? Eplain your answer. 0.09t. Population Growth: The population of a town increases according to the model P = 500e, where t is the time in years, with the year 000 corresponding to t = 0 (e.g. This year is 0). (a) Graph the function for the years 000 through 05. (b) Approimate the population in 05 and 05 using the graph. (c) Verify your values in (b) using the model (i.e. evaluate the function at those times). Section. - Logarithmic Functions & Graphs. Write the logarithmic equation in eponential form. (a) log8 = (b) log = (c) ln = Write the eponential equation in logarithmic form. (a) 9 = 7 (b) (c) e = f ( ) = log 6 at =. Evaluate the function at the value of without using a calculator:. Solve the equation. (a) log = (b) log = log Sketch the graph of each function. Make sure to show the -intercept and vertical asymptote. (a) g( ) = log6 (b) g( ) = log ( ) (c) f ( ) = log ( + ) Match the function to the graph. (a) (b) (c) (d) 6. f ( ) = log + 7. f ( ) = log 8. f ( ) = log ( + ) 9. f ( ) = log ( ) 0. Simplify using properties of natural logarithms. (a) ln e (b) ln.8 e (c) 0 7ln e. Home Mortgage: The model t = 6.65ln, if > 750, approimates the length of a home 750 mortgage of $50,000 at 6% in terms of monthly payments. In the model, t is the length of the mortgage in years and is the monthly payment in dollars. (a) Use the model to approimate the length of this mortgage when the monthly payment is $ Approimate the total monthly payments over the term of this mortgage. (b) Use the model to approimate the length of this mortgage when the monthly payment is $659.. Approimate the total monthly payments for this option. (c) What can you conclude from (a) and (b) regarding these payment options?

12 Section. - Logarithmic Properties. Rewrite the logarithm as a ratio using the change of base formula. Use both the common and natural logarithm bases. (a) log (b) loga 5 (c) log y. Use properties of logs to simplify the following epressions. 6 (a) log ( ) (b) 6 ln (c) ln e ( 5e ). 6. Use properties of logs to epand the following epressions. y. log0. ln y 5. ln t Use properties of logarithms to condense the following epressions. 7. ln 8 + 5ln 8. ln ln y ln z ln + ln + 5 ln( 5) 0. ( ). Simplify to an eact value without a calculator. (a) log 6 (b) ( ) 6 6. log b + 9. [ ln ln( + ) ln( ) ] log (c) log + log (d) ln e 5 5 y z ln e 6 5. Students participating in a psychology eperiment attended several lectures and were given an eam. Every month for the net year, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model 0 ( ) f ( t) = 90 5log t + 0 t where t is the time in months from the first test. (a) Graph the function over the given domain. You may use a calculator to help find points. (b) Find the average score on the original eam (t = 0). (c) Find the average score after 6 months. (d) Find the average score after months. (e) Find the time when the average score has dropped to 75. Section. - Solving Eponential & Logarithmic Equations Solve each equation. Leave each answer in eact form.. 7 = 9. ln ln = 0. e = 0. ln( + 5) = 8 5. log = 6. ( ) e = 8. + e = 9. log0 6 =. log0 ( 6) = + =. ( ). log log ( 8) log log + = 0 0 = 55 = 75 + e. The demand equation for an ipad is given by p = e where is the number of ipads + sold the demand (in thousands of units) and p is the price per unit. (a) Find the demand when the price is $600 per unit. (b) Find the price when the demand is 500,000 units ( = 500) 5. The yield V (in millions of cubic feet of timber per acre) for a forest at age t years is given by 8./ t V = 6.7e. (a) Find the yield after 0 years. (b) Find the time needed to obtain a yield of. million cubic feet.

13 Section.5 - Eponential and Logarithmic Models Match the function with its graph. (a) (b) (c). y = e /. y = 6e / y = 6 + log ( + ). 0 (d) (e) (f). y = e ( ) / 5 5. y = ln( + ) 6. y = + e 7. Consider an initial investment of $0,000 earning interest at 0.5% compounded continuously. (a) Find the amount of time needed to double the investment. (b) Find the interest rate required to double the investment in years. 8. Find the eponential model y b = ae that fits the following points. (0,) and (,0) 9. The number N of bacteria in a culture is given by the model N = 50e kt, where t is the time (hours). If N = 80 and t = 0, estimate the time required for the population to double in size. 0. The amount Y of yeast in a culture is given by the model 66 Y =, 0 t 8 where t represents time (hours). 0.57t + 7e (a) Make a table of values and graph this function over the given domain. (b) Use the model to predict the population for the 9 th hour (e.g. t =9) and the 0 th hour. (c) Find the limiting value of the population described by this model. (d) Eplain why the population of yeast follows a logistic growth model instead of an eponential decay model.. The Richter scale measures the magnitude R of an earthquake of intensity I using the model I R = log0, where I 0 =. I 0 (a) Find the intensity of the March, 0 earthquake off the coast of Japan (R = 9.0). (b) Find the intensity of the February 7, 8 earthquake near New Madrid, Missouri (R 7.0). (c) Find the magnitude of an earthquake with an intensity of 5, 00.. An F-6 fighter jet has a sound level of about 90 decibels (db) when landing, while the newer F-5 jet lands with a sound level of 05 db. The level of sound β (in decibels) is related to sound intensity I using the model β = 0log 0( I / I0), where I 0 = 0 - watts per m, which is roughly the faintest sound that can be heard by the human ear. Calculate the difference in sound intensity represented by this 5 db difference in sound level.

14 Chapter Review. Determine the eponential function whose graph is shown in the figure. y = ae b Determine if the following are an eponential growth function or an eponential decay function.. y e = +. y = + (5 ) Graph the function and analyze it for domain, range, and asymptotes.. y = log( ) + 5. y = + 5 Evaluate the logarithmic epression without using a calculator. 6. log 7. log 8 8. Rewrite the equation in eponential form. 0. log = 5. log Epand the following: log 0 9. = y ln e 7. ln y z =. Condense the following: y log z =. ln8 + 5ln z = 5. [ln( ) ln( )] ln( ) + + = Solve the equation = 7. e = = 9. ln = log = 6. = 5. log + = 7. log = e = 5. log( + ) + log( ) = 6. ln( + 5) ln( + ) = ln

15 7. Find the amount A accumulated after investing $50 for years at an interest rate of.6% compounded annually. 8. Find the amount A accumulated after investing $800 for 7 years at an interest of rate 6.% compouned quarterly. 9. How long would it take for your investment to double if it is compounded continuously at 8.5% interest rate? 0. If Jane invests $500 in a savings account with a 6% interest rate compounded monthly, how long will it take until Jane s amount has a balance of $500?. The time t in years for the world populations to double if it is increasing at a continuous rate of r is given by ln t =. Efficiently complete the table below and interpret your results. Use a graphing utility to graph the r function. r t. The number of bacteria in a culture is given by the model N = 50e kt where t is the time (in hours). If N = 80 when t = 0, estimate the time required for the population to double in size. Verify your estimate graphically.. The sales S (in thousands of units) of a cleaning solution after hundred dollars is spent on advertising are k given by S = 0( e ). When $500 is spent on advertising,,500 units are sold. Complete the model by solving for k. Estimate the number of units that will be sold if advertising ependitures are raised to $700.. The time (in hours per week) a student uses a math lab roughly follows the normal distribution ( 5.) / 0.5 y = e, 7, where is the time spent in the lab. Use a graphing utility to graph the function. From the graph, estimate the average time a student spends per week in the math lab The amount Y of yeast in a culture is given by the model Y =, 0 t 8, t in hours. 0.57t + 7e Use a graphing utility to graph the model. Use the model to predict the population for the 9 th hour and the 0 th hour. According to this model, what is the limiting value of the population? Why do you think the population of yeast follows a logistic growth model instead of an eponential growth model? 6. ( ) is equivalent to: a. b. 9 6 c. 9 9 d. 7 6 e In the real numbers, what is the solution of the equation 8 + = -? a. -/ b. -/ c. -/8 d. 0 e. /7

16 8. If log = p and log 5 = q, which of the following epressions is equal to 0? p+q b. p + q c. 9 p+q d. pq e. p + q 9. Whenever, y, and z are positive real numbers, which of the following epressions is equivalent to log + ½ log 6 y log z? a. y log ( ) z d. log ( z) log6 ( y ) b. log + log6 ( y ) z + e. log ( z) y + log 6 log z + log c. 6 y MORE EQUATION PRACTICE:. log ( + ) = log8. log 5( + ) = log5. log = log. log = log 9 5. log5 = log5 6. log = log 7 7. log ( ) + log = 5 8. log ( + ) log 9 = 9. log + log( + 5) = 0. log + log ( ) =. ln( + ) ln =. = 0. =. 8 =. 5. =.5 6. = 7. = = (0.5) 0. (0.) =.7.. π + = e. 5( ) 8 8. =. π = e = ( ) = 0.

17 Chapter : Trigonometry Essential Questions What is trigonometry? How are trigonometric functions similar to other functions we have studied? How are they different? What is the Unit Circle? Why do we study the unit circle? Why do certain strategies help you to verify trigonometric identities? How are the solutions to trigonometric equations similar to other equations we have studied? How are they different? Learning Targets Solve any or right triangle. Define all si trigonometric functions. Determine all of the values of the si trigonometric functions on the unit circle. Determine the values of the si trigonometric functions for any point in the Cartesian coordinate plane. Analyze and graph any trigonometric function given its equation and vice versa. Determine whether a set of data can be properly modeled using a trigonometric function. Section. - Right Triangle Trigonometry. Find the eact value of the si trigonometric functions of the angle θ. (a) (b). Sketch a right triangle corresponding to the trigonometric function. Find the eact value of the other five trigonometric functions. 7 (a) cotθ = 5 (b) cosθ = (c) cscθ = For problems -, find the indicated trigonometric function values using the given function values sin 60 =, cos60 = : (a) tan 60 (b) sin 0 (c) cos0 (d) cot 60 cscθ =, secθ = : (a) sinθ (b) cosθ (c) tanθ (d) sec( 90 θ ) cos α = : (a) secα (b) sinα (c) cotα (d) sin ( 90 α )

18 For problems 6-9, use trigonometric identities to transform one side of the equation into the other. 6. cscθ tanθ = secθ 7. cotθ sinθ = cosθ 8. tanθ + cotθ = csc θ tanθ 9. θ + θ θ = θ (csc cot )( cos ) sin 0. Mr. Wu stands 65 meters from the base of the Jin Mao Building in Shanghai, China. He looks up at Mr. Coulson, who on the 88 th floor observation deck, and estimates that the angle of elevation from the street to the top of the 88 th floor is 80. Calculate the approimate height of the building, and the distance between Mr. Wu and Mr. Coulson at this point.. In traveling across eastern Colorado you see the front range of the Rockies directly in front of you. For some reason, you stop the car and measure the angle of elevation to the closest peak to be.5. After you drive due west another miles, the urge again seizes you to stop and measure the angle of elevation to this same peak. Now it is 9. Using this information, find the approimate height of that mountain. Section. - Angles, Radians & Degrees. Determine the quadrant in which each angle lies. Then sketch in standard position. (a) π π (b) 57.5 (c).5 (d) 60.5 (e) 6. Determine two coterminal angles in (one positive, one negative) for each angle. (a) (b) (c). Find (if possible) the complement and supplement of each angle. π π (a) θ = 6 (b) θ = (c) θ = (d) θ = 8 (e) θ =. Rewrite each angle in either degrees or radians (eact value, without calculator). 7π (a) θ = 5 (b) θ = 70 (c) θ = (d) θ = π 6 5. Find the angle in radians. (a) (b)

19 6. Find the length of the arc on a circle with a radius of 9 feet and a central angle of A car is moving at a rate of 0 miles per hour, and the diameter of its wheels is.5 feet. (a) Find the linear speed of the tires in feet per minute. (b) Find the number of revolutions per minute the wheels are rotating. (c) Find the angular speed of the wheels in radians per minute. Section. - Unit Circle. Determine the eact values of the si trigonometric functions of the angle θ. (a) (b). Find the point on the unit circle that corresponds to the angle θ. π 5π (a) θ = (b) θ = (c) θ = π. Find the sine, cosine and tangent for the angle θ. π π π (a) θ = (b) θ = (c) θ = 6. Find the si trigonometric functions for the angle θ. 5π (a) θ = (b) θ = 6 5. Find the value of the given trigonometric function. (a) cos7π (b) 9π sin π (c) 9π sin 6 6. For the given trigonometric function, find the value of the indicated function. (a) Given cosθ =, find cos( θ) (b) Given sin( θ) =, find csc( θ ) 8 7. A bocce ball suspended from a Slinky bobs up and down, but because of friction the ball moves up and down less with each cycle. This is called damped harmonic motion, and in this case the vertical position of t the ball y (in feet) is given by the function y( t) = e cos6t, where t is the elapsed time (in seconds). Find the position of the ball at the following points in time. (a) t = 0 (b) t = (c) t =

20 Section. - Trig Functions for Any Circle. Determine the eact values of the si trigonometric functions for the angle θ. (a) (b). Find the values for the si trigonometric functions of θ. (a) cosθ = 5 and θ lies in Quadrant II (b) cscθ = and cotθ < 0. Find the reference angle θ for the angle θ. Sketch both angles in standard position. π (a) θ = 5 (b) θ = (c) θ = 95 (d) θ =.7. Find the indicated trigonometric value from the given function value and quadrant. (a) Find sinθ when cotθ = and θ lies in Quadrant II (b) Find cotθ when cscθ = and θ lies in Quadrant IV 9 (c) Find tanθ when secθ = and θ lies in Quadrant III 5 (d) Find cscθ when tanθ = and θ lies in Quadrant IV 5. Find the two solutions for each trigonometric equation on the interval 0 θ < π. (a) cscθ = (b) cotθ = 6. A company that produces water skis forecasts monthly sales over a two-year period using the following model: πt S =.+ 0.t +.sin where S is sales in thousands of units and t is time in months. 6 (a) If t = represents January of 0, estimate sales for this month. (b) Using the same basis, estimate the sales for June 0. (c) Eplain how this model accounts for the seasonal nature of water ski sales. Section.5 - Sine and Cosine Functions. Find the period and amplitude of each function. (a) y = cos (b) y = sin (c). Describe how f() and g() differ (consider amplitudes, periods and shifts). (a) f ( ) = sin, g( ) = sin( ) (b) f ( ) = cos, g( ) = cos (c) f ( ) = cos, g( ) = cos( + π ) (d) f ( ) = cos, g( ) = 6 + cos. Sketch the graphs of f() and g() on the same coordinate plane. Show two full periods. π y = cos (a) f ( ) = sin, g( ) = sin (b) f ( ) = cos, g( ) = cos π. Sketch the graph of the function by hand. Show two full periods. π (a) y = sinπ (b) y = 6cos + (c) π y = sin π π (d) y = cos

21 5. Mr. Coulson s resting blood pressure P (measured in mm Hg) is modeled by 8π P = 80 0cos t where t is the time (in seconds). (a) Graph the model. (b) Given that one period is equal to one heartbeat, find the time required for one heartbeat. (c) From the time per heartbeat, find Mr. Coulson s resting pulse rate in heartbeats per minute. (d) Find Mr. Coulson s maimum and minimum resting blood pressures. Section.6 - Secant and Cosecant Functions For problems -, match the function with its graph. Find the period of the function. (a) (b) (c). sec y =. π y = sec. y = csc For problems -9, sketch the graph of the function. Show two full periods.. y = sec 5. y = cscπ 6. y = sec + 7. csc y sec π y = csc π y = 8. = ( + ) 9. ( ) Section.6 - Tangent and Cotangent Functions For problems -, match the function with its graph. Find the period of the function. (a) (b) (c) π. y = tan. y = tan. y = cot For problems -9, sketch the graph of the function. Show two full periods.. y = cot 5. y = tan 6. y = cotπ 7. = cot y ( π ) π π y = tan + y = tan π

22 Chapter Review. For a right triangle, tanθ =. Find the value of the other five trig functions.. For a right triangle, cscθ =. Find the value of the si trig functions.. Name the quadrant that the angle lies in given that sec > 0, and sin < 0.. Name the quadrant that the angle lies in given that csc > 0, and tan < Suppose for an angle θ, cot θ = 7/ and sec θ < 0. Find the eact value of sin θ. 6. Convert 75 to radian measure. 7. Convert -7π/ to its eact degree measure and determine the quadrant of the terminal side of the angle. 8. Find the complement and supplement of π/5. 9. Find the reference angle for θ = -π/. Find the eact values of the si trigonometric functions of θ=π/. 0. If tan θ = / and sin θ < 0, find the quadrant of θ and the eact values of the trigonometric functions of θ.. If sec θ = -5/ and tan θ < 0, find the quadrant of θ and the eact values of the trigonometric functions of θ.. Identify the phase shift (horizontal shift), amplitude, and period of y = sin( π).. Find the amplitude and period of the sinusoidal graph given, then write an equation of the graph.. Given that sec =, find cot. 5. An escalator 5 feet in length rises to a platform and makes a 7 angle with the ground. Find the height of the platform. 6. A man at the top of a ramp 0 feet in length looks down to the end of the ramp which rises to a loading platform feet off the ground. For a safe ramp, the tanθ 0.5. Is this ramp safe? 7. cos 0 = 8. cot π/6 = 9. sin 0 = 0. tan π/ =. sec 0 =. csc 5π/ =. Use the fundamental identities to determine the simplified form of the epression. a) cos θ csc θ = b) tan θ cot θ = c) sin θ cot θ =. Sketch one period of the graph of f ( ) = sin ( π ) +. Find the period, amplitude, and phase shift.

23 π 5. Sketch one period of the graph of f ( ) = tan ( + ). Find the period and phase shift. π 6. Sketch one period of the graph of f ( ) = sec ( ). Find the period and phase shift. 7. Mr Coulson s oven doesn t bake cookies well, and he suspects the temperature controller is off. The repairman suggests that when the oven is set at 50 F, the temperature should fluctuate no more than ± 0 F during a 7 minute cookie-baking cycle, and that it follows a sinusoidal path. What would the equation for this temperature variation look like? Graph the epected equation over the course of the hour that Mr. C bakes cookies. Sketch a second graph of possible test data that would confirm Mr. C s suspicion. 8. How can a right triangle have negative values for trigonometric functions? 9. What is the measure of the reference angle for an angle of 00? a. 0 b. 60 c. 0 d. 50 e The legs of a right triangle measures 7 inches and 5 inches, respectively. What is the cosine of the triangle s smallest interior angle? 7 a. b c. d. e The figure below shows the path of a certain projectile launched from the ground at an angle of θ. The horizontal range, r, of this projectile when launched from the ground at a speed of 0 meters per second is modeled by r = 0 sin (θ). For this model, the angle measure θ that results in the greatest horizontal range, r, is 5 because: a. sinθ is greater than sin θ b. sin 90 is as large as sine can get. c. sin 5 is as large as sine can get. d. sin 5 is about e. sin θ is greater than sin θ.. The domain of the function y() = cos(5 ) + is all real numbers. Which of the following is the range of the function y()? a. y( ) b. y( ) c. y( ) d. y( ) e. All real numbers

24 Chapter 6: Law of Sines and Cosines Essential Questions Why do we have an ambiguous case? How do you solve a triangle when it is not right? How you determine which method to use (law of sines or law of cosines)? Learning Targets: Solve non-right triangles. Determine how many triangles are formed in the ambiguous case. Know when and how to solve the ambiguous case. Section 6. Law of Sines Solve the triangle(s) or write none if a triangle cannot be formed.. C = 05, c = 0, B = 0. C = 5, c = 5, B=0. A = 60, a =9, c = 0. A =., C = 5.6, c = A = 0, a = 5, b = A = 76, a =, b = 7. A = 58, a =.5, b =.8 8. B = 0, a = 9, c = 0 Section 6. Law of Cosines Solve the triangle.. a = 90, b =, c =. C = 08, a = 0, b = 7. a = 5, b = 0, c = 7. a =., b = 0.75, c =.5 5. A = 50, b = 5, c = 0 6. The baseball player in center field is playing approimately 0 feet from the television camera that is behind home plate. A batter hits a fly ball that goes to the wall 0 feet from the camera. The camera turns 6 to follow the play. Approimate the distance the center fielder has to run to make the catch.

25 Chapter 6 Review Solve the triangle(s) or write none if no triangle can be formed.. A = 6, a = 0, b =. A = 95, C = 5, c = 8. A = 5, a = 6, b = 7. A = 7, B = 5, c = 5. a =, c = 5, B = 5 6. a =, b = 8, c = 7 7. a =5, b =, c = A = 60, b = 50, c = 8 8. In ABCbelow, A measures 0, Bmeasures 5, and the length of BCis 65 meters. To the nearest meter, what is the length of AC? a. b. 5 c. 6 d. 7 e In the figure below, a radar screen shows ships. Ship A is located at a distance of 0 nautical miles and bearing 70, and Ship B is located at a distance of 0 nautical miles and bearing of 00. Which of the following is an epression for the straight-line distance, in nautical miles, between the ships? a. b. c. d. e (0)(0) cos (0)(0) cos (0)(0) cos (0)(0) cos (0)(0) cos 70