Why is the arctan of - square root of 3 equal a negative value (-pi/3) While the Arccos of -1/2 equal a positive value (2 pi/3)? Since it's the same concept, I'm not sure why the one value is negative while the other is positive. Thanks for your explanation!!

Answer 1

You have to know your CAST rules, in other words, you have to know in which quadrants each of the trig functions are positive or negative.

let's take the first one,
arctan(-√3)
we know that the tangent is negative in II and IV
your calculator probably gave you -60º if your setting is degrees.
Isn't that angle in the fourth quadrant ? (300º in a positive rotation)

another answer could have been 180-60 or 120º (test it, tan 120 = -√3)

Calculators have been programmed in the arc functions to provide the closest angle to zero ,(-60º is closer to 0 than the coterminal angle of 300º)

I hope you realize that -60º = -pi/3 radians.

for the arccos(-1/2)
we know that cos 120º = -1/2
and cos 240º = -1/2
(120º = 2pi/3 and 240º = 4pi/3)

Again, which angle would be closer to zero? It would be 120º or 2pi/3, which is the answer your calculator gave you.

BTW, the other answer to arccos(-1/2) would have been 4pi/3