The sign of a trigonometric function is dependent on the signs of the coordinates of the points on the terminal side of the angle. By knowing in which quadrant the terminal side of an angle lies, you also know the signs of all the trigonometric functions. There are eight regions in which the terminal side of an angle may lie: in any of the four quadrants, or along the axes in either the positive or negative direction (the quadrantal angles). Each situation means something different for the signs of the trigonometric functions.

The distance from a point to the origin is always
positive, but the signs of the
*x*
and
*y*
coordinates may be positive or negative.
Thus, in the first quadrant, where
*x*
and
*y*
coordinates are all positive, all six
trigonometric functions have positive values. In the second quadrant, only
sine and cosecant (the reciprocal of sine) are positive. In the third
quadrant, only tangent and cotangent are positive. Finally, in the
fourth quadrant, only cosine and secant are positive. The following
diagram may help clarify.

Figure %: The signs of the functions in the four quadrants

When an angle lies along an axis, the values of the trigonometric functions are either 0, 1, -1, or undefined. When the value of a trigonometric function is undefined, it means that the ratio for that given function involved division by zero. Below is a table with the values of the functions for quadrantal angles.

Figure %: The values of quadrantal angles

The points at which the values of a function are undefined are technically not
in the domain of that function. Therefore, the domain of sine and cosine is
all real numbers. The domain of tangent and secant is all real numbers except
+ *kΠ*
, where
*k*
is an integer. The domain of cosecant and
cotangent is all real numbers except
*kΠ*
, where
*k*
is an integer.

Take a Study Break!