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The Inverse Tangent Function - Problem 1

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Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

We’re talking about the inverse tangent function. Now remember that inverse tangent is the inverse of the restricted tangent function, y equals tangent x for x between negative pi over 2 and pi over 2. I think we’re ready for a definition.

Y equals inverse tangent x means, x equals tangent y for y between negative pi over 2 and pi over 2. When you think about inverse tangent, think about the angle whose tangent is x. so for instance, y equals inverse tangent of x, you’re looking for the angle whose tangent is x and the angle has to be between negative pi over 2 an pi over 2. I’ll show you what I mean with an example.

Let’s look at inverse tangent of 1. Think about the angle between negative pi over 2 and pi over 2 whose tangent is 1. You should know some of these basic values like tangent of pi over 4 equals 1. This value is between negative pI over 2 and pi over 2 and it’s the only one that has a tangent of 1 so this is our answer. Inverse tangent of 1 is pi over 4.

How about inverse tangent of root 3? We have to find a value of x whose tangent is negative root 3 and you can think about values, also remember that tangent is an odd function. So if you happen to remember that tangent of p over 3 is root, then the tangent of negative pi over3 is negative root 3. Tangent of negative I over 3 is negative root 3 and that means that inverse tangent of negative root 3 is negative pi over 3.

Inverse tangent of zero. Well tangent of zero equals zero so inverse tangent of zero equal zero. Inverse tangent of this zero equals this zero. Now remember there are lots of values lots of angles whose tangent is zero, zero, pi, 2 pi, 3 pi, any integer multiple of pi. The inverse tangent of zero is only zero. That’s inverse tangent in a nutshell. Remember inverse tangent is going to give you an angle all the inverse trig functions give you an angle and the input is a number. For inverse tangent the input is any number and the output is always an angle between negative pi over 2 and pi over 2.

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