Trigonometry Lecture Notes_part1-1

We will concentrate on the graph of the basic cosine

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, we will concentrate on the graph of the basic cosine curve on the interval [0, 2 π ]. The rest of the graph is made up of repetitions of this portion. Below is a graph of the cosine curve. From the graph we can see the following things: The domain is the set of all real numbers. The range consists of all numbers from [-1, 1] The period is 2 π . This function is an even function which can be seen from a graph by observing symmetry with respect to the y-axis.
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Note the similarity between the graphs of Sine and Cosine. In fact, they are related by the following identity: cos sin 2 x x π = + Like the sine graph, the graph of y = A cos B x has amplitude = | A | & period = 2 π /B . Example 27 Determine the amplitude and period of 3cos 2 y x π = - . Then graph the function for [-4, 4]. Steps 1. Determine the amplitude and the period using the form cos y A Bx = . 2. Divide the period by four and find the five key points using 1 4 i i period x x - = + . 3. Get your resulting y-values after plugging in the above x-values. 4. Extend the graph as you wish. The Graph of y = Acos(Bx - C)
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The graph of y = A cos ( Bx C ) is obtained by horizontally shifting the graph of y = A cos Bx so that the starting point of the cycle is shifted from x = 0 to x = C B . The number C B is called the phase shift . Amplitude = | A | & the period = 2 π /B . Remember! In the example above, C is not negative . In the general form, there is subtraction in the parentheses. Therefore, C is positive , which makes the function's phase shift to the right . Example 28 Determine the amplitude, period, and phase shift of ( ) 1 cos 4 2 y x π = + . Then graph one period of the function. Vertical shifts of the Sinusoidal Graphs Consider the following forms: ( ) sin y A Bx C D = - + and ( ) cos y A Bx C D = - + The constant D causes vertical shifts in the graphs of the functions. This will change the maximums and minimums so that the maximum becomes D A + and the minimum becomes D A - . Example 29 Graph one period of 1 cos 1 2 y x = - Section 6.6 Graphs of Other Trigonometric Functions The Graph of y = tanx The graph of tangent is very different from the graphs of sine and cosine. Here are some properties of the tangent function we should consider before graphing:
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The period is π . It is only necessary to graph tangent over an interval of length π . After this the graph just repeats. The tangent function is an odd function: Tan(-x) = -Tanx. The graph is therefore symmetric with respect to the origin. The tangent function is undefined at 2 π . Therefore, the graph will have a vertical asymptote at x = 2 π . We obtain the graph of y = tanx using some points on the graph and origin symmetry. The table below lists some tangent values over the interval 0, 2 π . X 0 6 π 4 π 3 π 5 12 π ( ) 75 ringoperator 17 36 π ( ) 85 ringoperator ( ) 89 89 180 π ringoperator 1.57 2 π Y = tan x 0 3 0.6 3 1 3 1.7 3.7 11.4 57.3 1255.
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