To do this, we will use a proof by contradiction. Since any number u can be written as 2 x for some x i. This is a contradiction. And the argument works for the other trigonometric functions as well. Thus, we get:.
The graphs of both functions are shown in Figure In general, the amplitude of a periodic curve f x is half the difference of the largest and smallest values that f x can take:.
In other words, the amplitude is the distance from either the top or bottom of the curve to the horizontal line that divides the curve in half, as in Figure 3. Not all periodic curves have an amplitude. For example, tan x has neither a maximum nor a minimum, so its amplitude is undefined. Likewise, cot x , csc x , and sec x do not have an amplitude. Since the amplitude involves vertical distances, it has no effect on the period of a function, and vice versa.
The graph is shown in Figure It just shifts the entire graph upward by 2. 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.
By multiplying the x in a trigonometric function by a constant, you can shorten or lengthen its period. Other factors you commonly see with trig functions include changes to the phase and amplitude, where the phase describes a change to the starting point on the graph, and amplitude is the function's maximum or minimum value, ignoring the negative sign on the minimum.
Chicago native John Papiewski has a physics degree and has been writing since He has contributed to "Foresight Update," a nanotechnology newsletter from the Foresight Institute. Related Articles What is a Periodic Function? What is the Period of Sine Function? How to Calculate the Phase Shift. 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. Graph of the tangent function. The stretching factor is A. Identify the stretching factor, A. Solution First, we identify A and B. Figure 2. Figure 3. There is no amplitude. Plot any three reference points and draw the graph through these points. Solution Step 1. Figure 4. How To: Given the graph of a tangent function, identify horizontal and vertical stretches.
Find the period P from the spacing between successive vertical asymptotes or x -intercepts. Determine a convenient point x , f x on the given graph and use it to determine A.
Figure 5. Solution The graph has the shape of a tangent function. To find the vertical stretch A , we can use the point 2,2. Try It 3 Find a formula for the function in Figure 6. Figure 6. Using the Graphs of Trigonometric Functions to Solve Real-World Problems Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. Find and interpret the stretching factor and period. Graph on the interval [0, 5]. Figure 7. Figure 8. Figure 9. Figure Sketch the asymptotes.
Plot any two reference points and draw the graph through these points. The graph for this function is shown in Figure Analysis of the Solution 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.
The cotangent function. Plot any two reference points. Step 8. The graph is shown in Figure One period of a modified cotangent function. Graph on the interval[0,5]. Find and interpret the stretching factor, period, and asymptote. Evaluate f 1 and f 2. Graph d x on this domain. Round to the second decimal place. What is the minimum distance between the fisherman and the boat?
When does this occur?
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