**The Cosecant Function**

The sine, cosine, and tangent functions form the cornerstone for the whole branch of practical mathematics commonly called trigonometry. However, three more circular functions exist. Their values represent the reciprocals of the values of the preceding three functions. To understand the definitions of these functions, look again at Fig. 1-5.

**Fig. 1-5** . The unit circle is the basis for the trigonometric functions.

Imagine the ray *OP,* subtending an angle *θ* with respect to the *x* axis, and emanating out 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 state this simple equation:

csc *θ* = 1/ *y* _{0}

This function is the reciprocal of the sine function. That is to say, 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 equal to 1/0. But we can’t do anything with a quotient in which the denominator is 0. (Resist the temptation to call it “infinity”!)

**The Secant Function**

**Fig. 1-5** . The unit circle is the basis for the trigonometric functions.

Keeping the same vision in mind, consider 1/x _{0} . This is defined as 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. That is, as long as cos *θ* is not equal to zero:

sec *θ* = 1/(cos *θ* )

The secant function is not defined for 90° (π/2 rad), or for any odd multiple thereof.

**The Cotangent Function**

**Fig. 1-5** . The unit circle is the basis for the trigonometric functions.

There’s one more circular function to go. You can guess it by elimination: *x* _{0} / *y* _{0} . It is called the *cotangent* function, abbreviated 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 the reciprocal of the tangent function:

cot *θ* = 1/tan *θ*

This function, like the tangent function, “blows up” 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).

**Conventional Angles**

Once in a while you will hear or read about an angle whose measure is negative, or whose measure is 360° (2π rad) or more. In trigonometry, any such angle can always be reduced to something that is at least 0° (0 rad), but less than 360° (2π rad). If you look at Fig. 1-5 one more time, you should be able to see why this is true. Even if the ray *OP* makes more than one complete revolution counterclockwise from the *x* axis, or if it turns clockwise instead, its orientation can always be defined by some counterclockwise angle of least 0° (0 rad) but less than 360° (2π rad) relative to the *x* axis.

Any angle ø of the non-standard sort, like 730° or –9π/4 rad, can be reduced to an angle *θ* that is at least 0° (0 rad) but less than 360° (2π rad) by adding or subtracting some whole-number multiple of 360° (2π rad).

Multiple revolutions of objects, while not usually significant in pure trigonometry, are sometimes important in physics and engineering. We don’t have to worry about whether a vector pointing along the positive *y* axis has undergone 0.25, 1.25, or 101.25 revolutions counterclockwise, or 0.75, 2.75, or 202.75 revolutions clockwise. But scientists must sometimes deal with things like this, and when that happens, non-standard angles such as 36,450° must be expressed in that form.

**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 an electronic calculator! Most personal computers have a calculator program built into the operating system. You might have to dig around in the operating system folders to find it, but once you do, you can put a shortcut to it on your computer’s desktop. Use 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 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

Practice Problems for these concepts can be found at: The Circle Model Practice Test