What is Arctan?Date: 8/28/96 at 11:15:0 From: Anonymous Subject: What is Arctan? I don't understand a question. it says: Let y = f(x) = sin (arctan x). Is sin the same as arctan X? Is it something different? Do I multiply the two together? Thank You! Date: 10/17/96 at 10:58:10 From: Doctor Roger Subject: Re: What is Arctan? The sine and the arc-tangent are different functions, and both require arguments to make sense (argument means the input to the function). In your equation the argument of the Arctan is (x), while the argument of the sin is (Arctan x). They are *not* multiplied together. The notation means to take the arctangent of x first, then find the sine of the result. In order to simplify the equation, first consider arctan x. It represents that angle whose tangent is x. For example, if theta = arctan(x), then x = tan(theta). Let's make a diagram of a triangle having an angle theta, such that the tangent of theta is x: .| . | . | H . | x . | . | . theta | --------------- 1 Notice that the Pythagorean Theorem tells us that the hypotenuse H is given by H = sqrt (1+x^2). Thus we have: f(x) = sin (arctan (x)) = sin (theta) = x / H = x / sqrt(1+x^2) You might object that the "size" of my triangle was irrelevant - and that is right! I could just as easily have picked a triangle with sides of 2x and 2, or any other side lengths that are in the ratio of x:1. (Remember - the tangent of theta has to be x.) The hypotenuse would have scaled accordingly, and we would have come up with the same expression for sin(arctan(x)). (Try it!) -Doctor Roger, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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