![]() ![]() ![]() f ( x ) = sin x does not pass the Horizontal Line Test and must be restricted to find its inverse. Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test. Inverse Sine Function Inverse Sine Function Sin x has an inverse function on this interval.When x0 in II y<0 in IV Range Domain arctan(x) arccos(x) arcsin(x) For the inverse trig functions the ratio is the input and the angle is the output. Remember, the angle is the input for a trig function and the ratio is the output. The table below will summarize the parameters we have so far.y=tan(x) y=arctan(x) Like the sine function, the domain of the section of the tangent that generates the arctan is The other trig functions require similar restrictions on their domains in order to generate an inverse.Since the domain and range for the section are the domain and range for the inverse cosine are This section includes all outputs from –1 to 1 and all inputs in the first and second quadrants. The chosen section for the cosine is in the red frame.Consider the cosine function: What do you think would be a good domain restriction for the cosine? Congratulations if you realized that the restriction we used on the sine is not going to work on the cosine. The other inverse trig functions are generated by using similar restrictions on the domain of the trig function.Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians). In the tradition of inverse functions then we have: The thing to remember is that for the trig function the input is the angle and the output is the ratio, but for the inverse trig function the input is the ratio and the output is the angle.Note how each point on the original graph gets “reflected” onto the graph of the inverse.The domain of the chosen section of the sine is So the range of the arcsin is The range of the chosen section of the sine is so the domain of the arcsin is. To get a good look at the graph of the inverse function, we will “turn the tables” on the sine function. The new table generates the graph of the inverse.I have plotted the special angles on the curve and the table.Lets zoom in and look at some key points in this section. Quadrant I angles generate the positive ratios and negative angles in Quadrant IV generate the negative ratios. We are going to build the inverse function from this section of the sine curve because: This section picks up all the outputs of the sine from –1 to 1. Take a look at the piece of the graph in the red frame.But we can come up with a valid inverse function if we restrict the domain as we did with the previous function. You can see right away that the sine function does not pass the horizontal line test. A similar restriction on the domain is necessary to create an inverse function for each trig function.The graph now passes the horizontal line test and we do have an inverse: Note how each graph reflects across the line y = x onto its inverse. ![]() So how is it that we arrange for this function to have an inverse? We consider only one half of the graph: x > 0.f(2) = 4 and f(-2) = 4 so what is an inverse function supposed to do with 4? By definition, a function cannot generate two different outputs for the same input, so the sad truth is that this function, as is, does not have an inverse. Consider the graph of Note the two points on the graph and also on the line y=4.Do you know what is wrong? Congratulations if you guessed that the top function does not really have an inverse because it is not 1-1 and therefore, the graph will not pass the horizontal line test. But something is not quite right with this pair. Let us begin with a simple question: What is the first pair of inverse functions that pop into YOUR mind? This may not be your pair but this is a famous pair.Round towards positive infinityĪs opposed to the floor function, ceil(x) rounds the elements of x to the nearest integers towards infinity. It is most useful while you want to round the result of an operation.įloor(x) rounds the elements of x to the nearest integers towards minus infinity. This function will allow you to round values towards the nearest integer if you are not interested in working with decimal numbers. We will use as an example the remainder of the division of 5 by 2, which is one. This function helps you find the remainder after a division. Matlab code sqrt(a*b) Remainder after division Matlab code atan(3/2) e Matlab (exponential) Matlab code acos(0.5) Trigonometric inverse tangent (arctan) Matlab code asin(1) Trigonometric inverse cosine (arccos) Matlab code tan(3*pi/2) Trigonometric inverse sine (arcsine) Matlab code cos(pi) Trigonometric tangent ![]()
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