Question

What quadrants in the unit circle is the sin, cos, tan and all real numbers

Answer 1

Sine and Cosine are defined over every real number, really over every complex number if you want to go there. So the answer is never.

Pardon me but this seems like a slightly confused question.

When we talk about sinθ , the θ is an angle. θ is just a real number that’s used in the common parameterization of the unit circle,

(x,y)=(cosθ,sinθ)

θ is interpreted as the angle between two rays, one the positive x axis, and one the ray originating at the origin and intersecting the unit circle at (x,y). The angle is given by the arc length of the unit circle cut by the two rays.

There are other ways to parameterize the circle, the most important being

(x,y)=(1−t21+t2,2t1+t2)

which is on the unit circle because of the easily verifiable identity known to Euclid, (1−t2)2+(2t)2=(1+t2)2

The parameterization is defined for all real t but doesn’t quite get the entire unit circle. It’s missing (−1,0). We can allow t=∞ , essentially treating t as a projective parameter, a ratio, and get the entire circle.