6-3: Graphing Sine And Cosine Functions
The average monthly temperatures for a citydemonstrate a repetitious behavior. For cities in the Northernp li c a tiHemisphere, the average monthly temperatures are usually lowestin January and highest in July. The graph below shows the average monthlytemperatures ( F) for Baltimore, Maryland, and Asheville, North Carolina, withJanuary represented by 1.METEOROLOGYonAp Use the graphsof the sine andcosine functions.l WorealdOBJECTIVEGraphing Sine andCosine FunctionsR6-3yBaltimore807060Temperature 5040( F)302010OAsheville24681012141618202224 tMonth 6Model for Baltimore’s temperature: y 54.4 22.5 sin (t 4)Model for Asheville’s temperature: 6y 54.5 18.5 sin (t 4)In these equations, t denotes the month with January represented by t 1.What is the average temperature for each city for month 13?Which city has the greater fluctuation in temperature?These problems will be solved in Example 5.Each year, the graph for Baltimore will be about the same. This is also truefor Asheville. If the values of a function are the same for each given interval of thedomain (in this case, 12 months or 1 year), the function is said to be periodic.The interval is the period of the function.PeriodicFunction andPeriodA function is periodic if, for some real number , f (x ) f (x) for each xin the domain of f.The least positive value of for which f (x) f (x ) is the period of thefunction.Lesson 6-3Graphing Sine and Cosine Functions359
Example1 Determine if each function is periodic. If so, state the period.a.The values of the function repeat foreach interval of 4 units. The function isperiodic, and the period is 4.y42Ob.246812 x10The values of the function do not repeat.The function is not periodic.y42O246812 x10Consider the sine function. First evaluate y sin x for domain values between 2 and 2 in multiples of .4xsin x7 3 5 2 402 22142 2 3 0 2 42 4 2 1 220 3 02 212 2 1 2 20 4245 423 27 422 0To graph y sin x, plot the coordinate pairs from the table and connect themto form a smooth curve. Notice that the range values for the domain interval 2 x 0 (shown in red) repeat for the domain interval between 0 x 2 (shown in blue). The sine function is a periodic function.yy sin x 2 1O 2 x 1By studying the graph and its repeating pattern, you can determine thefollowing properties of the graph of the sine function.Propertiesof the Graphof y sin x360Chapter 61.2.3.4.5.6.The period is 2 .The domain is the set of real numbers.The range is the set of real numbers between 1 and 1, inclusive.The x-intercepts are located at n, where n is an integer.The y-intercept is 0. The maximum values are y 1 and occur when x 2 n,2where n is an integer.3 7. The minimum values are y 1 and occur when x 2 n,2where n is an integer.Graphs of Trigonometric Functions
Examples9 2 Find sin 2 by referring to the graph of the sine function.9 9 Because the period of the sine function is 2 and 2 , rewrite as a sum22involving 2 .9 4 22 2 2 2 (2) This is a form of 2 n.9 2 2So, sin sin or 1.3 Find the values of for which sin 0 is true.Since sin 0 indicates the x-intercepts of the function, sin 0 if n ,where n is any integer.4 Graph y sin x for 3 x 5 .The graph crosses the x-axis at 3 , 4 , and 5 . It has its maximum value9 7 of 1 at x , and its minimum value of 1 at x .22Use this information to sketch the graph.yy sin x1O3 4 5 x 1l WoreaAponldR5 METEOROLOGY Refer to the application at the beginning of the lesson.p li c a tia. What is the average temperature for each city for month 13?Month 13 is January of the second year. To find the average temperature ofthis month, substitute this value into each equation.BaltimoreAsheville y 54.4 22.5 sin (t 4)6y 54.5 18.5 sin (t 4) y 54.4 22.5 sin (13 4)63 2 6 y 54.5 18.5 sin (13 4)63 2y 54.4 22.5 sin y 54.5 18.5 sin y 54.4 22.5( 1)y 31.9y 54.5 18.5( 1)y 36.0In January, the average temperature for Baltimore is 31.9 , andthe average temperature for Asheville is 36.0 .b. Which city has the greater fluctuation in temperature?Explain.The average temperature for January is lower in Baltimorethan in Asheville. The average temperature for July is higher inBaltimore than in Asheville. Therefore, there is a greaterfluctuation in temperature in Baltimore than in Asheville.Lesson 6-3Graphing Sine and Cosine Functions361
Now, consider the graph of y cos x.xcos x7 3 5 2 412 2 3 0 1 2 2 02 212 2204242244 3 425 4 1 2 20 223 7 2 02 2124yy cos x1 2 O2 x 1By studying the graph and its repeating pattern, you can determine thefollowing properties of the graph of the cosine function.Propertiesof the Graphof y cos xExample1.2.3.4.5.6.The period is 2 .The domain is the set of real numbers.The range is the set of real numbers between 1 and 1, inclusive. The x-intercepts are located at n, where n is an integer.2The y-intercept is 1.The maximum values are y 1 and occur when x n, where n is aneven integer.7. The minimum values are y 1 and occur when x n, where n is anodd integer.6 Determine whether the graph represents y sin x, y cos x, or neither.y1 9 8 7 Ox 1The maximum value of 1 occurs when x 8 .maximum of 1 whenx n cos xThe minimum value of 1 occurs at 9 and 7 . minimum of 1 whenx n cos x17 215 2The x-intercepts are and .These are characteristics of the cosine function. The graph is y cos x.362Chapter 6Graphs of Trigonometric Functions
C HECKCommunicatingMathematicsFORU N D E R S TA N D I N GRead and study the lesson to answer each question.1. CounterexampleSketch the graph of a periodic function that is neither thesine nor cosine function. State the period of the function.2. Name three values of x that would result in the maximum value for y sin x.3. Explain why the cosine function is a periodic function.4. MathJournal Draw the graphs for the sine function and the cosine function.Compare and contrast the two graphs.Guided Practicey5. Determine if the function is periodic. If so, state2the period.O2468x 2Find each value by referring to the graph of the sine or the cosine function. 6. cos 25 7. sin 28. Find the values of for which sin 1 is true.Graph each function for the given interval.9. y cos x, 5 x10. y sin x, 4 7 2 xy11. Determine whether the graph representsy sin x, y cos x, or neither. Explain.1O4 5 6 7 x 112. Meteorology 6The equation y 49 28 sin (t 4) models the averagemonthly temperature for Omaha, Nebraska. In this equation, t denotesthe number of months with January represented by 1. Compare the averagemonthly temperature for April and October.E XERCISESPracticeDetermine if each function is periodic. If so state the period.A13.yO14.2468 10 12 xy 22 4O16. y x 5 17. y x 2www.amc.glencoe.com/self check quiz15.4y42468xO204060 x118. y xLesson 6-3 Graphing Sine and Cosine Functions363
Find each value by referring to the graph of the sine or the cosine function.19. cos 8 3 22. sin 220. sin 11 21. cos 27 23. sin 224. cos ( 3 )25. What is the value of sin cos ?26. Find the value of sin 2 cos 2 .Find the values of for which each equation is true.B27. cos 128. sin 129. cos 030. Under what conditions does cos 1?Graph each function for the given interval.31. y sin x, 5 x 3 32. y cos x, 8 x10 33. y cos x, 5 x 3 9 34. y sin x, 2x13 27 35. y cos x, 2x 3 27 36. y sin x, 2x11 2Determine whether each graph is y sin x, y cos x, or neither. Explain.37.y38.1O4 5 1C6 xyOy39.32117 8 7 x9 6 Ox 5 140. Describe a transformation that would change the graph of the sine function tothe graph of the cosine function.41. Name any lines of symmetry for the graph of y sin x.42. Name any lines of symmetry for the graph of y cos x.43. Use the graph of the sine function to find the values of for which eachstatement is true.a. csc 1b. csc 1c. csc is undefined.44. Use the graph of the cosine function to find the values of for which eachstatement is true.a. sec 1GraphingCalculator364b. sec 1c. sec is undefined.Use a graphing calculator to graph the sine and cosine functions on the same setof axes for 0 x 2 . Use the graphs to find the values of x, if any, for whicheach of the following is true.45. sin x cos x46. sin x47. sin x cos x 148. sin x cos x49. sin x cos x 150. sin x cos x 0Chapter 6 Graphs of Trigonometric Functionscos x0
l WoreaAponldRApplicationsand ProblemSolvingp li c a ti51. Meteorology 6The equation y 43 31 sin (t 4) models the averagemonthly temperatures for Minneapolis, Minnesota. In this equation, t denotesthe number of months with January represented by 1.a. What is the difference between the average monthly temperatures for Julyand January? What is the relationship between this difference and thecoefficient of the sine term?b. What is the sum of the average monthly temperatures for July and January?What is the relationship between this sum and value of constant term?Consider the graph of y 2 sin x.What are the x-intercepts of the graph?What is the maximum value of y?What is the minimum value of y?What is the period of the function?Graph the function.How does the 2 in the equation affect the graph?52. Critical Thinkinga.b.c.d.e.f.The equation P 100 20 sin 2 tmodels a person’s blood pressure P inmillimeters of mercury. In this equation, t istime in seconds. The blood pressure oscillates20 millimeters above and below 100 millimeters,which means that the person’s blood pressureis 120 over 80. This function has a period of1 second, which means that the person’s heartbeats 60 times a minute.a. Find the blood pressure at t 0, t 0.25,t 0.5, t 0.75, and t 1.b. During the first second, when was the bloodpressure at a maximum?c. During the first second, when was the bloodpressure at a minimum?53. MedicineyThe motion of a weight on aspring can be described by a modifiedcosine function. The weight suspendedfrom a spring is at its equilibrium pointwhen it is at rest. When pushed a certainOtdistance above the equilibrium point, theweight oscillates above and below theequilibrium point. The time that it takesfor the weight to oscillate from the highest point to the lowest point and back to54. Physics the highest point is its period. The equation v 3.5 cos t k models themvertical displacement v of the weight in relationship to the equilibrium point atany time t if it is initially pushed up 3.5 centimeters. In this equation, k is theelasticity of the spring and m is the mass of the weight.a. Suppose k 19.6 and m 1.99. Find the vertical displacement after0.9 second and after 1.7 seconds.b. When will the weight be at the equilibrium point for the first time?c. How long will it take the weight to complete one period?Lesson 6-3 Graphing Sine and Cosine Functions365
Consider the graph of y cos 2x.55. Critical Thinkinga. What are the x-intercepts of the graph?b. What is the maximum value of y?c. What is the minimum value of y?d. What is the period of the function?e. Sketch the graph.56. EcologyIn predator-prey relationships, the number of animals in eachcategory tends to vary periodically. A certain region has pumas as predatorsand deer as prey. The equation P 500 200 sin [0.4(t 2)] models thenumber of pumas after t years. The equation D 1500 400 sin (0.4t) modelsthe number of deer after t years. How many pumas and deer will there be in theregion for each value of t?a. t 0Mixed Reviewb. t 10c. t 2557. TechnologyA computer CD-ROM is rotating at 500 revolutions per minute.Write the angular velocity in radians per second. (Lesson 6-2)58. Change 1.5 radians to degree measure. (Lesson 6-1)59. Find the values of x in the interval 0 x 2 .360 for which sin x 2(Lesson 5-5)2xx2 4 . (Lesson 4-6)60. Solve x 22 xx2 461. Find the number of possible positive real zeros and the number of negativereal zeros of f(x) 2x 3 3x 2 11x 6. Then determine the rational roots.(Lesson 4-4)62. Use the Remainder Theorem to find the remainder when x3 2x2 9x 18is divided by x 1. State whether the binomial is a factor of the polynomial.(Lesson 4-3)63. Determine the equations of the vertical and horizontal asymptotes, if any, ofx2 . (Lesson 3-7)g(x) x2 x64. Use the graph of the parent function f(x) x 3 to describe the graph of therelated function g(x) 3x 3. (Lesson 3-2)65. Find the value of 24 11 10 . (Lesson 2-5) 34566. Use a reflection matrix to find the coordinates of the vertices of ABC reflectedover the y-axis for vertices A (3, 2), B (2, 4), and C (1, 6). (Lesson 2-4)367. Graph x y. (Lesson 1-3)268. SAT/ACT PracticeHow much less is the perimeterof square RSVW than the perimeter of rectangleRTUW?A 2 unitsB 4 unitsC 9 unitsD 12 unitsE 20 units366Chapter 6 Graphs of Trigonometric FunctionsRS 2TVU5WExtra Practice See p. A36.
ofMATHEMATICSFUNCTIONSMathematicians and statisticians usefunctions to express relationships amongsets of numbers. When you use aspreadsheet or a graphingcalculator, writing an expressionas a function is crucial forcalculating values in thespreadsheet or for graphingthe function.Early EvidenceIn about2000 B.C., the Babylonians usedthe idea of function in makingtables of values for n and n3 n2,for n 1, 2, , 30. Their workindicated that they believed they couldshow a correspondence between thesetwo sets of values. The following is anexample of a Babylonian table.nn3 n21221230?The RenaissanceModern Era The 1800s brought JosephLagrange’s idea of function. He limited themeaning of a function to a power series.An example of a power series isx x 2 x 3 , where the threedots indicate that the patterncontinues forever. In 1822, JeanFourier determined that anyfunction can be representedby a trigonometric series.Peter Gustav Dirichlet usedthe terminology y is a functionof x to mean that each firstelement in the set of ordered pairsis different. Variations of his definitionJohann Bernoullican be found in mathematics textbookstoday, including this one.In about 1637, RenéDescartes may have been the first person touse the term “function.” He defined a functionas a power of x, such as x 2 or x 3, where thepower was a positive integer. About 55 yearslater, Gottfried von Leibniz defined afunction as anything that related to a curve,such as a point on a curve or the slope of acurve. In 1718, Johann Bernoulli thoughtof a function as a relationship between avariable and some constants. Later in thatsame century, Leonhard Euler’s notion of afunction was an equation or formula withvariables and constants. Euler also expandedthe notion of function to include not only thewritten expression, but the graphicalrepresentation of the relationship as well.He is credited with the modern standardnotation for function, f(x).Georg Cantor and others working in thelate 1800s and early 1900s are credited withextending the concept of function fromordered pairs of numbers to ordered pairsof elements.Today engineers like Julia Chang usefunctions to calculate the efficiency ofequipment used in manufacturing. She alsouses functions to determine the amount ofhazardous chemicals generated during themanufacturing process. She uses spreadsheetsto find many values of these functions.ACTIVITIES1. Make a table of values for the Babylonianfunction, f(n) n3 n2. Use values of nfrom 1 to 30, inclusive. Then, graph thisfunction using paper and pencil, graphingsoftware, or a graphing calculator.Describe the graph.2. Research other functions used by notablemathematicians mentioned in this article.You may choose to explore trigonometricseries.3.Find out more aboutpersonalities referenced in this article andothers who contributed to the history of functions. Visit www.amc.glencoe.comHistory of Mathematics367
Explain why the cosine function is a periodic function. 4. Math Journal Draw the graphs for the sine function and the cosine function. Compare and contrast the two graphs. 5. Determine if the function is periodic. If so, state the period. Find each value by referring to the graph of the sine or the cosine function. 6. cos 2 7. sin 5 2 8.
Continue Graphing Sine and Cosine (Period Changes) Worksheet graphing problems #9 – 16 on pp. 9 – 10 Thursday 10/24 Writing Equations of sine and cosine functions (Notes p. 11) Worksheets pp. 12 – 13 Friday 10/25 Writing functions cont’d Quiz - Graphing Sine
output generated: modified sine wave, and pure sine wave1. A modified sine wave can be seen as more of a square wave than a sine wave; it passes the high DC voltage for specified amounts of time so that the average power and rms voltage are the same as if it were a sine wave.
medium voltage cables. All VLF test units from Megger can also be used for sheath testing and sheath fault pin-pointing. TECHNICAL DATA VLF Sine 34 kV VLF Sine 45 kV VLF Sine 62 kV VLF test voltage 0 to 34 kV peak 0 to 45 kV peak 0 to 62 kV peak Frequency 0.01 to 0.1Hz 0.01 to 0.1Hz 0.01 to 0.1Hz Wave form Sine Sine Sine Testing cable capacitance
4. Advantages of a Pure Sine Wave Inverter Today s inverters come in two basic output waveforms: modified sine (which is actually a modified square wave) and pure sine wave. Modified sine wave inverters approxi
Graphing Sine and Cosine Functions Desmos Activity 1. Use your unit circle and fill in the exact values of the sine function for each of the following angles (measured in radians). sin0 2 sin C sinC sin 3C 2 sin2C 2. Use the values in the table above to graph one cycle of the sine curve on the coordinate plane below. 3.
Graphing Sine and Cosine functions: amplitude, phase shift, and vertical slide changes. Cofunctions . Complete Note sheets pp.2 to 5 . Tuesday . 10/11 . Graphing Sine and Cosine functions cont’d . period changes notes p. 6 : Worksheet graphing problems # 1– 8 . pp. 7 -8 . Wednesday . 10/12 . Periods 1-3: PSAT . Periods 5-7: more graphing!!
Lesson 8-4 Sine and Cosine Ratios 439 Sine and Cosine Ratios The tangent ratio, as you have seen, involves both legs of a right triangle.The sine and cosine ratios involve one leg and the hypotenuse. of &A of &A These equations can be abbreviated: sin A cos A Writing Sine and Cosine Ratios Use the triangle to write each ratio. a. sin T .
3.3 Sine wave filter 12 3.3.1 Symmetrical sine wave filter 13 3.3.2 Sinus plus symmetrical and asymmetrical sine wave filter 14 4. Sine wave filter selection 16 4.1 Current and voltage rating 16 4.2 Frequency 17 4.3 Require
