Graphs of Sine Cosine Funct - Section 4.5 Graphs of Sine...

Info icon This preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 4.5 Graphs of Sine and Cosine Functions 1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Intro to the Graph of Trigonometric Functions In this section, we use graphs of sine and cosine functions to visualize their properties. We use the traditional symbol x, rather than u or t, to represent the independent variable. We use the symbol y for the dependent variable, or the function’s value at x. Thus, we will be graphing y = sin x and y = cos x in rectangular coordinates. In all graphs of trigonometric functions, the independent variable, x, is measured in radians. 2
Image of page 2
The Graph of y 􏳁= sin x The trigonometric functions can be graphed in a rectangular coordinate system by plotting points whose coordinates satisfy the function. Thus, we graph y = sin x by listing some points on the graph. Because the period of the sine function is 2π, we will graph the function on the interval [0, 2π]. The rest of the graph is made up of repetitions of this portion. 3
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
4
Image of page 4
Continuation of the previous problem showing 3 cycles 5
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Properties of the graph of Sine The graph of y = sin x allows us to visualize some of the properties of the sine function. • The domain is (- ∞ , ∞), the set of all real numbers. The graph extends indefinitely to the left and to the right with no gaps or holes. • The range is [-1, 1], the set of all real numbers between -1 and 1, inclusive. The graph never rises above 1 or falls below -1. 6
Image of page 6
Properties of the graph of Sine The period is 2π. The graph’s pattern repeats in every interval of length 2π or 360. The function is an odd function : sin(- x) = - sin x. This can be seen by observing that the graph is symmetric with respect to the origin. 7
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Graphing Variations of y=sin x To graph variations of y = sin x by hand, it is helpful to find 1)The three x-intercepts (beginning, middle and end of its full period) 2)maximum points, (one maximum point ¼ of the way of its full period) 3)minimum points, (one minimum point ¾ of the way of its full period) Thus, key points in graphing sine functions are obtained by dividing the period into four equal parts. 8
Image of page 8
The graph of y = sin x forms the basis for graphing functions of the form y = A sin x. 9
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
10
Image of page 10
Graphing y=2 Sin x For example, consider y = 2 sin x, in which A = 2. We can obtain the graph of y = 2 sin x from that of y = sin x if we multiply each y-coordinate on the graph of y = sin x by 2. Figure 4.65 shows the graphs. The basic sine curve is stretched and ranges between -2 and 2, rather than between -1 and 1. However, both y = sin x and y = 2 sin x have a period of 2π. 11
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The amplitude of y = A sin x In general, the graph of y = A sin x ranges between – I A I and I A I. Thus, the range of the function is – I A I ≤ y ≤ I A I.
Image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern