Introduction to Secondary Circular Functions
The three functions defined above form the cornerstone for trigonometry. But three more circular functions exist. Their values represent the reciprocals of the values of the primary circular functions.
The Cosecant Function
Imagine the ray OP in Fig. 121, at an angle θ with respect to the x axis, pointing outward from the origin, and intersecting the unit circle at the point P = ( x _{0} , y _{0} ). The reciprocal of the ordinate, that is, 1/ y _{0} , is defined as the cosecant of the angle θ . The cosecant function is abbreviated csc, so we can say this in mathematical terms:
csc θ = 1/ y _{0}
The cosecant function is the reciprocal of the sine function. For any angle θ , the following equation is always true as long as sin θ is not equal to zero:
csc θ = 1 / sin θ
The cosecant function is not defined for 0° (0 rad), or for any multiple of 180° (π rad). This is because the sine of any such angle is equal to 0, which would mean that the cosecant would have to be 1/0. We can’t do anything with a quotient in which the denominator is 0.
Figure 126 is a graph of the cosecant function for values of θ between –3π and 3π. Note the angles at which the function “explodes.”
Fig. 126. Graph of the cosecant function for values of θ between –3π rad and 3π rad.
The Secant Function
Now consider the reciprocal of the abscissa, that is, 1/ x _{0} , in Fig. 121. This is the secant of the angle θ . The secant function is abbreviated sec, so we can define it like this:
sec θ = 1 / x _{0}
The secant of any angle is the reciprocal of the cosine of that angle. As long as cos θ is not equal to zero, the following is true:
sec θ = 1 / cos θ
The secant function is not defined for 90° (π/2 rad), or for any positive or negative odd multiple thereof. Figure 127 is a graph of the secant function for values of θ between –3π and 3π. Note the angles at which the function “blows up.”
Fig. 127. Graph of the secant function for values of θ between –3π rad and 3π rad.
The Cotangent Function
There’s one more circular function. Consider the value of x _{0} / y _{0} at the point P where the ray OP crosses the unit circle. This quotient is called the cotangent of the angle θ . The word “cotangent” is abbreviated as cot. For any ray anchored at the origin and crossing the unit circle at an angle θ :
cot θ = x _{0} / y _{0}
Because we already know that sin θ = y _{0} and cos θ = x _{0} , we can express the cotangent function in terms of the sine and the cosine:
cot θ = cos θ / sin θ
The cotangent function is also the reciprocal of the tangent function:
cot θ = 1 / tan θ
This function, like the tangent function, “explodes” at certain values of θ . Whenever y _{0} = 0, the denominator of either quotient above becomes zero, and the cotangent function is not defined. This occurs at all integer multiples of 180° (π rad). Figure 128 is a graph of the cotangent function for values of θ between –3π and 3π.
Fig. 128. Graph of the cotangent function for values of θ between –3π rad and 3π rad.

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