When drawn from the origin, a line 135 degrees counterclockwise from the positive x-axis lies in quadrant II. Sine is defined as the ratio between the opposite side of a triangle and the hypotenuse of the triangle. Cosecant is defined as the reciprocal of sine, so the sine and cosecant of an angle must have the same sign. A right triangle can be drawn in quadrant II composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The opposite side of this triangle lies above the x-axis, and is therefore represented by a positive number. Since the opposite side and hypotenuse are both represented by positive numbers, the fraction O/H is positive. Therefore, sin 135 is positive, as is its reciprocal, csc 135.

When drawn from the origin, a line 225 degrees counterclockwise from the positive x-axis lies in quadrant II. Tangent is defined as the ratio between the opposite and adjacent sides of a triangle. Cotangent is defined as the reciprocal of tangent, so the tangent and cotangent of an angle must have the same sign. A right triangle can be drawn in quadrant III composed of any point on that line, the origin and a point on the x-axis. The opposite side of this triangle lies below the x-axis, and is therefore represented by a negative number. The adjacent side of this triangle lies on the negative x-axis, and is therefore represented by a negative number. Since the opposite side and adjacent side have the same sign, the fraction O/A is positive. Therefore, tan 225 is positive, as is its reciprocal cot 225.

When drawn from the origin, a line 315 degrees counterclockwise from the positive x-axis lies in quadrant IV. Tangent is defined as the ratio between the opposite and adjacent sides of a triangle. A right triangle can be drawn in quadrant IV composed of any point on that line, the origin and a point on the x-axis. The opposite side of this triangle lies below the x-axis, and is therefore represented by a negative number. The adjacent side of this triangle lies on the positive x-axis, and is therefore positive. Since the opposite side and hypotenuse have opposite signs, the fraction O/A is negative. Since the triangle formed is a 45-45-90 triangle, the opposite and adjacent side lengths have the same absolute value, so the fraction O/A has an absolute value of 1. Therefore, tan 315 = -1.

Recall that . Since the ratio of any two real numbers is undefined when the denominator is equal to ,  must be undefined for those values of  where . Restricting our attention to those values of  between  and ,  when  or . Hence,  is undefined when  or .

The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Sine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (SOH, or sin x = opposite/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition sin x = opposite/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = sin x is all real numbers.

The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (CAH, or cos x = adjacent/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition cos x = adjacent/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = cos x is all real numbers.

Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos (pi/2) is 0, so the secant of (pi)/2 must be undefined.

Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (pi) is 0, so the cotangent of (pi) must be undefined.

Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Secant is the reciprocal of cosine, so the secant of any angle x for which cos x = 0 must be undefined, since it would have a denominator equal to 0. The value of cos 3(pi/2) is 0, so the secant of 3(pi)/2 must be undefined.

Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cotangent is the reciprocal of tangent, so the cotangent of any angle x for which tan x = 0 must be undefined, since it would have a denominator equal to 0. The value of tan (0) is 0, so the cotangent of (0) must be undefined.