Trig Inverses: sin(arctan x)
Date: 11/04/2002 at 15:06:32 From: Erin Subject: Trig Inverses We got a question in my trig class today, and I don't know where to start. We discussed the inverses of trig functions, but when the teacher threw an x in there, I didn't understand. Here is the problem she gave us. sin(arctan x) Thanks for all your help.
Date: 11/05/2002 at 14:05:41 From: Doctor Ian Subject: Re: Trig Inverses Hi Erin, When you have an expression like sin(x) it means that if you supply the value of an angle, the function will return a corresponding value: sin(45 degrees) = sin(pi/4 radians) = sqrt(2)/2 So if you have sin( f(x) ) it implifes that f(x) is yet another kind of function, that takes some value of x and returns an angle, which will then be the argument to the sine function. Does that make sense so far? Well, arctan is just such a function. It takes a value, such as 1, and returns the angle that _would_ give you that value, if used as an input to the tangent function. So 1) tan(45 deg) = 1 2) arctan(1) = 45 deg 3) tan( arctan(1) ) = tan( 45 deg ) = 1 4) arctan( tan( 45 deg) ) = arctan(1) = 45 deg The other common name for 'arctan' is 'inverse tangent', and these examples show why: the tan() and arctan() functions cancel each other out (although you have to be careful to keep track of what quadrant you're talking about). The expression sin(arctan x) means give me some value; arctan will tell you for which angle tan(x) = that value; and sin(...) will tell you the sine of the angle. Let's work through it for the following triangle, |\ a | \ c |__\ b A using x = (a/b) for our argument: sin( arctan (a/b) ) = sin( A ) because tan(A) = a/b, so arctan(a/b) = A = a/c because sin(A) = a/c Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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