Explain why tan^-1 3 is defined, but cos^-1 3 is undefined.?

cos^(-1) (3) is undefined because there does not exist any angle such that cos (angle) = 3.

However, tan^(-1) 3 exists because there is an angle such that tan (angle) = 3.

The tangent function of an angle, tan A = sin A / cos A, is defined for all angles, except for those angles whose cosine is equal to zero, because we can't divide by zero. However, as we approach one of these angles, the function takes on an infinitely large (+ infinity), or infinitely small (- infinity) value. Here's a link:

http://search.yahoo.com/search;_ylt=Awrj4i6RAHhRrX...

So, the function is said to be defined for all real number values. In particular tan^-1 (3) ~ 71.565 degrees. To confirm it, enter 71.565 into a calculator (make sure it is in degree mode first), and then hit the tan button. It will return a value very close to 3. Clear the result. Then enter 3; hit the shift button, and then hit tan-1. It will return a value close to 71.565.

The cosine function fluctuates between values of -1 and +1, so it can never take on the value of +3. Therefore, cos^-1 (3) (read as the angle whose cosine is equal to 3) does not exist. So cos^-1 (3) is undefined.

When you do cos(x), no matter what x is, the output is always somewhere between -1 and 1. The inverse cosine function which you are writing cos^-1 just reverses the cosine function. 3 is outside the range of the cosine function, so there is no way to "reverse" it. Hence, it is undefined.

For the tangent function, the range is all of the real numbers. So for example, tan(1.570) is over 1255. This means that for any real number, including 3, we can reverse the tangent function. Inverse tangent is defined for all real numbers.

"Defined" means the function can take the value in question and spit out an answer.