Section 7.1 Graphs of Sine and Cosine

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1 Section 7.1 Graphs of Sine and Cosine OBJECTIVE 1: Understanding the Graph of the Sine Function and its Properties In Chapter 7, we will use a rectangular coordinate system for a different purpose. We will plot points and label them (, ) ybut the value (or independent variable ) will represent an angle measured in radians and the y value (or dependent variable) will represent a trigonometric function of the angle, either y = sin, y = cos, y = tan, y = csc, y = sec, or y = cot. For eample, consider the function y = sin. When the graph of y = sin 1 =, y = sin =. Thus, the ordered pair 2. 1, $ represents a point that lies on % 2 Complete the tables below and plot the ordered pairs to create a graph of y = sin, from 0 to y = sin y = sin 5 7 y = sin 2 5 y = sin Draw a set of aes. Determine appropriate tic marks for and y, plot points, and connect them with a smooth curve.

2 Definition Periodic Function A function f is said to be periodic if there is a positive number P such that f ( + P) = f () for all in the domain of f. The smallest number P for which f is periodic is called the period of f. Use the information below to assist you in determining the period of f () = sin. f % ( = f $ ' 5 9 % 2 ) = f % ( = f % ( It appears that the period of the periodic function f () = sin is. ( = Using information above and the graph of f () = sin, write the characteristics of the Sine Function: The domain is (interval notation) The range is (interval notation) The function is periodic with a period of P = The y-intercept is The -intercepts or zeros are of the form where n is The function is (odd or even or neither), which means sin( ) =. The graph is symmetric about. The function obtains a relative maimum at = where n is an. The maimum value is The function obtains a relative minimum at = where n is an. The minimum value is Draw the graph of f () = sin with domain 2, Using the graph of y = sin list all values of on the interval 11, % $ 2 ' that satisfy the ordered 1 $ pair,. 2 %

3 7.1.7 Use the periodic property of y = sin to determine which of the following epressions is y = 1 2 sin 1 % $ ' (, 5 + i. sin $ equivalent to ) *, - y ii. sin 2 $ iii. sin 29 $ iv. sin 1 % $ ' %. % % (, 0) y = 2cos$ 5 % ' OBJECTIVE 2: Understanding the Graph of the Cosine Function and its Properties Complete the tables below and plot the ordered pairs to create a graph of y = cos, from 0 to y = cos y = cos 5 7 y = cos 2 5 y = cos Draw a set of aes. Determine appropriate tic marks for and y, plot points, and connect them with a smooth curve. Use the information below to assist you in determining the period of f () = cos. f % ( = f $ ' 5 9 % 2 ) = f % ( = f % ( It appears that the period of the periodic function f () = cos is. ( =

4 Using information above and the graph of f () = cos, write the characteristics of the Sine Function: The domain is (interval notation) The range is (interval notation) The function is periodic with a period of P = The y-intercept is The -intercepts or zeros are of the form where n is The function is (odd or even or neither), which means cos( ) =. The graph is symmetric about. The function obtains a relative maimum at = where n is an. The maimum value is The function obtains a relative minimum at = where n is an. The minimum value is Draw the graph of f () = cos with domain 2, Using the graph of y = cos list all values of on the interval, 5 $ pair (, 0). % ' that satisfy the ordered Use the periodic property of y = cos to determine which of the following epressions is equivalent to cos 25 $. % i. cos 5 $ % ii. cos 7 $ iii. cos 1 $ % % iv. cos 7 % $ ' We focus on sketching five points on every graph, then using knowledge of the sine function and cosine function to draw the graph. These five points evenly divide one cycle of the sine or cosine curve into fourths, or quarters and well be called the five quarter points.

5 List the 5 quarter points of y = sin and show them on a graph of the function. List the 5 quarter points of y = cos and show them on a graph of the function. OBJECTIVE : Sketching Graphs of the Form y = Asin and y = Acos There are four factors that affect the graph of a sine or cosine curve. The first is amplitude. Definition The amplitude of a sine or cosine curve is the measure of half the distance between the maimum and minimum values. Draw a graph and indicate the amplitude. Trigonometric functions of the form y = Asin and y = Acos have an amplitude of A and a range of [ A, A ] Determine the amplitude and range of the function y = 5cos and use the quarter points to sketch its graph. OBJECTIVE : Sketching Graphs of the Form y = sinb and y = cosb The second factor that affects a sine or cosine curve is a change in period. Functions of the form y = sinb and y = cosb have a period other than 2 when B 1.

6 The period of a sine or cosine curve is equal to P = 2 B where B > 0. Find the quarter points by subdividing the interval 0, 2 $ % B ' ( into four equal subintervals of length 2 B Given that B > 0, since y = sin is an (odd or even) function, sin( B) = and the graph of y = sin( B) is (how related to y = sin(b)) Given that B > 0, since y = cos is an (odd or even) function, cos( B) = and the graph of y = cos( B) is (how related to y = cos(b)) Determine the period of the function y = cos( ) and plot the quarter points to sketch its graph Determine the period of the function y = sin( ) and plot the quarter points to sketch its graph.

7 OBJECTIVE 5: Sketching Graphs of the Form y = Asin(B) and y = Acos(B) StepsforSketchingFunctionsoftheForm y = Asin( B) and y Acos( B) = Step1:If B < 0,usetheevenandoddpropertiesofthesineandcosinefunctiontorewritethefunction inanequivalentformsuchthat B > 0 WenowusethisnewformtodetermineAandB. Step2:Determinetheamplitudeandrange.Theamplitudeis A.Therangeis $ A, A %. 2 Step:Determinetheperiod.Theperiodis P =. B 2 Step: Anintervalforonecompletecycleis 0, % B $.Subdividethisintervalintoequalsubintervals oflength 2 2 bystartingwith0andadding $ % B B ' tothecoordinateofeachsuccessivequarter point. Step5:Multiplytheycoordinatesofthequarterpointsof y = sin or y = cos bya%todetermine theycoordinatesofthecorrespondingquarterpointsforthenewgraph. Step:Connectthequarterpointstoobtainonecompletecycle Determine the amplitude, range, and period of the function y = 1 2 sin $ 1 % ' and sketch its graph.

8 7.1. Determine the amplitude, range, and period of the function y = 2cos$ 5 % ' and sketch its graph.

Section 5.2 Graphs of the Sine and Cosine Functions

A Periodic Function and Its Period Section 5.2 Graphs of the Sine and Cosine Functions A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p) = f(x) for all x in