Question

Show how to solve the following; answer must be with fractions and pi

theta=tan^-1(-6/6)
I can get the decimal form from calculator, but I require the fractional answer and I need to know how it even gets obtained. How do we know what inverse tan of -6/6 will = to?

Answer 1

3pi/4 and -pi/4

Explanation:

We can simplify -6/6 to -1.

Therefore, this function can be simplified to arctan(-1).

Recall that the meaning of arctan is to find a value that will get the value inside the parenthesis when taken the tangent of it. In other words, tan(x) = -1.

Recall that tan(x) = sin(x)/cos(x). Now recall that sin(pi/4) and cos(pi/4) are both sqrt(2)/2, meaning that tan(pi/4) is 1. To make it -1, we can either make sin(x) -1 while keeping cos(x) 1, or the other way around.

If x is -pi/4, cos(x) will still be 1, but sin(x) will be -1, so tan(-pi/4) will be -1.

If x is 3pi/4, cos(x) will be -1, but sin(x) will still be 1, so tan(3pi/4) will be -1.

Side note: there are still infinite more answers. You can attain them by adding or subtracting 2pi as many times as you want from 3pi/4 or -pi/4 and still get an arctan of -1.