The Sine and Cosine Functions

Transcription

1 Lesson -5 Lesson -5 The Sine and Cosine Functions Vocabular BIG IDEA The values of cos and sin determine functions with equations = sin and = cos whose domain is the set of all real numbers. From the eact values of sines, cosines, and tangents ou calculated in Lesson -, ou can see the shape of a function called the. Activit Step The table below contains some eact values of sin. It also shows decimal approimations to those values. Complete the table, using a unit circle to help ou. Mental Math If gasoline costs $.00 a gallon and a car gets 5 miles to the gallon, what does it cost for gas per mile? (degrees) 0º 0º 5º 0º 90º 0º 5º 50º 80º (radians) 0 5 sin (eact) 0?????? sin (appro.) 0 0.5???? 0.707?? (degrees) 0º 5º 0º 70º 00º 5º 0º 0º (radians) 7 5??? sin (eact)??????? sin (appro.)???? 0.8??? Step Here is a graph of the fi rst fi ve points in the fi rst part of the table. Cop this graph, and on it plot the points ou found in Step. Then draw a smooth curve through the points (continued on net page) The Sine and Cosine Functions 7

2 Chapter Step Check Step b using a graphing utilit to plot = sin for 0º 0º and for 0. The Graph of the Sine Function The function that maps each real number to the -coordinate of the image of (, 0) under a rotation of is called the. From the unit circle, ou can tell that sin is positive when 0º < < 80º and negative when 80º< < 0º. The maimum value is, when = 90º, and the minimum value is, when = 70º. (, 0) (0, ) (0, ) (cos, sin ) (, 0) QY Restate the preceding paragraph for in radians. A graph of the, for 0º 0º, is shown at the right. To make it easier to locate zeros, maima, and minima, the scale on the horizontal ais is in multiples of and 90º. This is one ccle of the graph of the. Because the image of (, 0) under a rotation of repeats itself ever radians, the -coordinates in the ordered pairs of the function f with equation f() = sin repeat ever. Thus, the graph above can be easil etended both to the right and left without calculating an new sine values. The graph of the entire has infinitel man ccles. A graph showing three complete ccles of the appears below. f () = sin Notice from the graph that the -intercept of the is 0. The s -intercepts (zeros) are...,,, 0,,,,,..., that is, the integer multiples of. 8 Trigonometric Functions

3 Lesson -5 As the graph on the previous page makes clear, the domain of the is the set of real numbers. Because the maimum and minimum values of the are and (the -intercepts of the unit circle) the range is the interval. Also notice that the graph of the is point-smmetric about the origin. Thus, the is an odd function. This is because of the Opposites Theorem that states for all, sin( ) = sin. The Graph of the Cosine Function Remember that the image of (, 0) under a rotation of magnitude is (cos, sin ). The function that maps each real number to the first coordinate of the image of (, 0) under a rotation of is called the cosine function. The has man characteristics like those of the. A graph of the is shown below. f () = cos (, 0) (0, ) (0, ) (cos, sin ) (, 0) Activit Use the defi nitions and graphs of the sine and s to fi ll in the table. (degrees) (radians) (degrees) (radians) Domain???? Range???? Zeros???? Maima sin = when = 90º, 50º, 80º,??? Minima???? Questions COVERING THE IDEAS. a. Identif the domain and the range of the. b. Find five values of such that sin = 0.. a. Sketch a graph of = sin for 0. b. Find all values of on this interval such that sin =. c. Find all values of on this interval for which sin = 0.5. The Sine and Cosine Functions 9

4 Chapter. a. Cop the table below. Fill in eact and approimate values (rounded to three decimal places) for some of the coordinates of points on the graph of the. 0 cos (eact) b. Use the points from the table to graph = cos.. a. Sketch a graph of = cos for 0º 70º. b. Find five values of on this interval for which cos = Describe three was in which the graph of = cos is like the graph of = sin and two was in which the graphs are different. APPLYING THE MATHEMATICS. Describe the translation with the smallest positive magnitude that maps the graph of g() = cos onto that of = sin. 7. The graph of the is reflection-smmetric over the line with equation =. a. What propert of sines is a result of this smmetr? b. Name two other lines of smmetr for the graph. 8. In a stable environment, predator-pre populations can be modeled b sine waves. Refer to the graph below. 5 0?? cos (appro.) ?? cos (eact)???????? _ cos (appro.)???????? Population Pre Predator Months a. Describe what is happening with the pre population when the predator population is at its peak. b. Describe what is happening with the pre population when the predators are the fewest. 50 Trigonometric Functions

5 Lesson Use the graph of = f() at the right. Suppose f is known to be either the or the. a. Evaluate f ( _ ). b. For what value of, in the interval from 0 to, does f() =? c. Tell whether f is the or. Justif our answer. 0. The graph of the is reflection-smmetric to the -ais. What propert of cosines is a result of this smmetr? 0 REVIEW In and, A is a point on a circle with center at the origin. Find the coordinates of A for the given value of. (Lesson -).. A 0 = (, 0) 0 7 = (, 0) A. In radians, what is the sum of the measures of the angles of a pentagon? (Lesson ). An old 78 RPM record revolves through 78 revolutions in a minute. How man radians is this per second? (Lesson ) 5. The measure of an angle is k radians. Convert this measure to degrees. (Lesson ). The students in Ms. T. Chare s st period geometr class measured their heights h in centimeters and recorded the following five-number summar of their data: h = 5; min = 7; Q = 5; median = 8; Q = 7; ma = 88. Are there an outliers in the data set? Eplain our answer. (Lesson -) EXPLORATION 7. At what angle to the -ais does the graph of = sin pass through (0, 0)? Give numerical and visual evidence supporting our answer. QY ANSWER From the unit circle, ou can tell that sin is positive when 0 < < and negative when < <. The maimum value is, when =, and the minimum value is, when =. The Sine and Cosine Functions 5