First, let's convert pi/3 to degrees. We do this with the conversion factor of 180/pi
pi/3 xx 180/pi = 60^@
Now we have to calculate cos60^@. By the special triangle of sides 1, sqrt(3) and 2 we have that cos60^@ = 1/2
We can state the new expression as being tan^-1(-2 xx 1/2) = tan^-1(-1)
We must calculate the tan^-1(-1). We can do this by first looking in which quadrants tan will be negative. Tan is negative in quadrants II and IV.
Furthermore, for the ratio to be -1/1, the 45^@, 45^@, 90^@; 1, 1, sqrt(2) special triangle will be implicated.
By reference angles, we have that tan^-1(-1) = 180 - 45^@ and 360 - 45^@, so tan^-1(-1) = 135^@ and 315^@.
Depending on how your teacher wants your answer tan^-1(-2cos(pi/3)) = 135^@ and 315^@ or tan^-1(-2cos(pi/3)) = (3pi)/4 and (7pi)/4
Hopefully this helps!
