quadrantsis it really important to say in what quadrant the angle is?
Yest it is important. When we know that location of the angle, we can determine the values of the trigonometric identities.
We will compare the angle pi/3 into the first, second, third, and 4th quadrants.
If pi/3 is in the 1st quadrant:
==> cos(pi/3) = 0.5
If pi/3 is in the 2nd quadrant:
==> pi/3 = pi - pi/3 = 2pi/3
==> cos (2pi/3) = - 0.5
If pi/3 is in the 3rd quadrant.
==> pi/3 = pi + pi/3 = 4pi/3
==> cos(4pi/3) = - 0.5
If pi/3 is in the 4th quadrant:
==> pi/3 = 2pi - pi/3 = 5pi/3
==> cos( 5pi/3) = 0.5
Then, we notice that the sign changes as the angle changes the location.
Of course it is important!
1/2 and -1/2 are not representing the same amount.
We know that the functions sine, cosine, tangent and cotangent, have positive and negative values, depending on the quadrant they are located.
Let's have an example.
The value of sine function of the angle of 30 degrees is 1/2 in the first and the second quadrant. The value of sine function of the angle of 30 degrees, in the 3rd and 4th quadrants, is negative and it is -1/2.
For this reason, it is important to establish the quadrant where the angle is located, before calculating the value of trigonometric function required.