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Viewed 4k times. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. The interval over which the sine wave repeats itself is called the period. Like any wave, the shapes have recognizable features such as peaks high points and troughs low points.
After this point, the cycle repeats indefinitely, producing the same features and values as the angle increases in the positive x direction. The sine function takes the value of zero at zero degrees, where as the cosine is 1 at the same point. You get the tangent function by dividing sine by cosine. Although it has x values at which it becomes undefined, the tangent function still has a definable period. The three other trig functions, cosecant, secant and cotangent, are the reciprocals of sine, cosine and tangent, respectively.
Although their graphs have undefined points, the periods for each of these functions is the same as for sine, cosine and tangent.
Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function. Steps 6—7. We can use two reference points, the local minimum at 0, 2. Figure 11 shows the graph. Step 6. There is a local minimum at 1. Figure 12 shows the graph.
The excluded points of the domain follow the vertical asymptotes. Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function. Figure 13 shows the graph.
What are the domain and range of this function? The vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval are shown by dots. Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.
We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following. Plot two reference points. Step 7. Step For the following exercises, rewrite each expression such that the argument x is positive. For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes. For the following exercises, find and graph two periods of the periodic function with the given stretching factor, A , period, and phase shift.
What is the function shown in the graph? Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let x , measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and x is measured negative to the left and positive to the right. A laser rangefinder is locked on a comet approaching Earth. A video camera is focused on a rocket on a launching pad 2 miles from the camera.
Skip to main content. Module 2: Periodic Functions. Search for:. Figure 1.
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