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!!
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
The arctan and arccos functions are both inverse trigonometric functions used to find the angle that corresponds to a specific trigonometric ratio. However, they have different ranges and behavior, which explains the difference in the signs of their outputs for certain inputs.
Let's break down the two cases you mentioned:
1. Arctan of -√3:
To calculate the arctan of -√3, we want to find the angle whose tangent is -√3. The tangent function is negative in the third and fourth quadrants of the unit circle. In these quadrants, the y-coordinate is negative, and the x-coordinate can be either positive or negative.
When we evaluate the arctan of -√3, we obtain -π/3. The negative sign indicates that the angle is in the negative direction or clockwise rotation from the positive x-axis. This is because in the third quadrant, the angles are measured in a clockwise direction from the positive x-axis.
Therefore, the negative sign signifies the specific quadrant in which the angle lies and represents the direction of rotation from the positive x-axis.
2. Arccos of -1/2:
Finding the arccos of -1/2 means finding the angle whose cosine is -1/2. The cosine function is negative in the second and third quadrants of the unit circle. In these quadrants, the x-coordinate is negative, and the y-coordinate can be either positive or negative.
When we evaluate the arccos of -1/2, we obtain 2π/3. The positive sign shows that the angle is in the positive direction or counterclockwise rotation from the positive x-axis. This is because in the second quadrant, the angles are measured in a counterclockwise direction from the positive x-axis.
So, the positive sign indicates the specific quadrant in which the angle lies and represents the counterclockwise rotation from the positive x-axis.
In summary, the sign of the output for inverse trigonometric functions represents the direction of rotation from the positive x-axis and indicates the specific quadrant in which the angle lies. It is important to understand the ranges and behavior of each function to correctly interpret the signs of their outputs.