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pre-calc 4.5 notes

# pre-calc 4.5 notes - Section 4.5 Graphs of Sine and Cosine...

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Section 4.5 Graphs of Sine and Cosine Functions In this section, we look at the graphs of sine and cosine functions. We use the traditional symbol x , rather than θ or t , to represent the independent variable. We use the symbol y for the dependent variable, or the function’s value at x . So we are looking at graphs of y = sin x and y = cos x in the rectangular coordinate system. In all graphs of trigonometric functions, the independent variable, x , is measured in radians. The Graph of y = sin x Graph y = sin x by listing some points on the graph. Since the period of sine is 2π, graph the function on [0, 2π] and the rest of the graph is made up of repetitions of this portion. x 0 6 π 3 π 2 π 3 2 π 6 5 π π 6 7 π 3 4 π 2 3 π 3 5 π 6 11 π y = sin x 0 2 1 2 3 1 2 3 2 1 0 2 1 - 2 3 - -1 2 3 - 2 1 - 0 You can get a more complete graph of y = sin x by continuing the portion shown above to the left and to the right. Properties of the sine function The domain is (-∞, ∞), the set of all real numbers. The range is [-1, 1], the set of all real numbers between -1 and 1, inclusive. The period is 2π. (The graph’s pattern repeats in every interval of length 2π.) The function is an odd function: sin(- x ) = - sin x Graphing Variations of y = sin x It is helpful to find x-intercepts, maximum points, and minimum points to graph these variations. The x- coordinates of the key points are x 1 = value of x where the cycle begins, x 2 = x 1 + 4 period , x 3 = x 2 + 4 period , x 4 = x

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