**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. 12-1, 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 12-6 is a graph of the cosecant function for values of *θ* between –3π and 3π. Note the angles at which the function “explodes.”

**Fig. 12-6.** 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. 12-1. 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 12-7 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. 12-7.** 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 12-8 is a graph of the cotangent function for values of *θ* between –3π and 3π.

**Fig. 12-8.** Graph of the cotangent function for values of *θ* between –3π rad and 3π rad.

**Values of Circular Functions**

Now that you know how the circular functions are defined, you might wonder how the values are calculated. The answer: with a calculator! Most personal computers have a calculator program built into the operating system. You might have to dig around in the folders to find it, but once you do, you can put a shortcut to it on your computer’s desktop. Put the calculator in the “scientific” mode.

The values of the sine and cosine function never get smaller than –1 or larger than 1. The values of other functions can vary wildly. Put a few numbers into your calculator and see what happens when you apply the circular functions to them. Pay attention to whether you’re using degrees or radians. When the value of a function “blows up” (the denominator in the unit-circle equation defining it becomes zero), you’ll get an error message on the calculator.

**Secondary Circular Functions Practice Problems**

**Practice 1**

Use a portable scientific calculator, or the calculator program in a personal computer, to find the values of all six circular functions of 66°. Round your answers off to three decimal places. If your calculator does not have keys for the cosecant (csc), secant (sec), or cotangent (cot) functions, first find the sine (sin), cosine (cos), and tangent (tan) respectively, then find the reciprocal, and finally round off your answer to three decimal places.

**Solution 1**

You should get the following results. Be sure your calculator is set to work with degrees, not radians.

sin 66° = 0.914

cos 66° = 0.407

tan 66° = 2.246

csc 66° = 1/(sin 66°) = 1.095

sec 66° = 1/(cos 66°) = 2.459

cot 66 = 1/(tan 66°) = 0.445

Find practice problems and solutions for these concepts at: Trigonometry Practice Test.

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From Everyday Math Demystified: A Self-Teaching Guide. Copyright © 2004 by The McGraw-Hill Companies, Inc. All Rights Reserved.