Graphs of Sine Cosine Funct

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

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Section 4.5 Graphs of Sine and Cosine Functions 1

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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
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

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Continuation of the previous problem showing 3 cycles 5

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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
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

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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
The graph of y = sin x forms the basis for graphing functions of the form y = A sin x. 9

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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

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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.
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