the line y = -4 passes through the points (0,-4) and (2,-4).
the line x = 0 passes through the points (0,0) and (0,-4).
these points are taken of the mechanically generated graph shown below:
the hypotenuse of the right triangle formed is on the line y = -2x.
they form a right triangle whose angle with the vertex at the point (0,0) has the side opposite to it with a length of 2 and a hypotenuse with a length of sqrt(2^2 + 4^2) = sqrt(20).
the vertex of the 90 degree angle of this right triangle is at the point (0,-4).
the sine of the angle with the vertex at the point (0,0) is equal to 2 / sqrt(20).
the angle is equal to the arcsine of (2/sqrt(20)) = 26.56505118 degrees.
that's the reference angle of the angle that is in the fourth quadrant.
the equivalent angle is iis equal to 360 - 26.56505118 = 333.4349488 degrees.
the sine, cosine, tangent, cotangent, secant, and cosecant of the reference angle are:
sine = .4472135955
cosine = .894427191
tangent = .5
cotangent = 2
secant = 1.118033989
cosecant = 2.236067977
the sine, cosine, tangent, cotangent, secant, and cosecant of the equivalent angle in the fourth quadrant are:
sine = -.4472135955
cosine = .894427191
tangent = -.5
cotangent = -2
secant = 1.118033989
cosecant = -2.236067977
as you can see, the trig functions are exactly the same except that the sine and tangent functions are negative when the angle is in the fourth quadrant, while the cosine function is positive.
the secant, cosecant, and cotangent functions always follows the sign of the cosine, sine, and tangent functions, since they are reciprocals of those functions.
if the sine is negative, the cosecant is also negative.
if the cosine is positive, the secant is also positive.
if the tangent is negative, the cotangent is also negative.
a picture of the graph that i drew is shown below:
A is the reference angle.
B is the equivalent angle in the fourth quadrant.