Cosecant, Secant, and Cotangent

Cosecant, Secant, and Cotangent

In this chapter we'll introduced three more trigonometric functions: the cosecant, the secant, and the cotangent. These functions are written as csc(), sec(), and cot() respectively. They are the functions defined by the formulas below:

csc()

=

1 sin()

sec()

=

1 cos()

cot()

=

cos() sin()

Graphs of cosecant, secant, and cotangent

0

0

274

Periods

Cosecant, secant, and cotangent are periodic functions. Cosecant and secant have the same period as sine and cosine do, namely 2. Cotangent has period , just as tangent does. In terms of formulas, the previous two sentences mean that

csc( + 2) = csc()

sec( + 2) = sec()

cot( + ) = cot()

It's easy to check why these functions have the periods that they do. For

example, because sine has period 2--that is, because sin( + 2) = sin()--

we can check that

1

1

csc( + 2) =

=

= csc()

sin( + 2) sin()

Similarly, the secant function has the same period, 2, as the function used to define it, cosine.

Even and odd

Recall that an even function is a function f (x) with the property that f (-x) = f (x). Examples include x2, x4, x6, and cosine.

We can add secant to the list of functions that we know are even functions. That is, sec(-) = sec(). The reason secant is even is that cosine is even:

1

1

sec(-) =

=

= sec()

cos(-) cos()

An odd function is a function f (x) with the property that f (-x) = -f (x). Examples include x3, x5, x7, sine, and tangent.

Cosecant and cotangent are odd functions, meaning that csc(-) = - csc() and cot(-) = - cot(). We can check that these identities are true by using that sine is an odd function and that cosine is even:

1

1

csc(-) =

=

= - csc()

sin(-) - sin()

cos(-) cos()

cot(-) =

=

= - cot()

sin(-) - sin()

275

Cofunction identities

Sine and cosine, secant and cosecant, tangent and cotangent; these pairs of functions satisfy a common identity that is sometimes called the cofunction identity :

sin

2

-

= cos()

sec

2

-

= csc()

tan

2

-

= cot()

These identities also "go the other way":

cos

2

-

= sin()

csc

2

-

= sec()

cot

2

-

= tan()

Let's check one of these six identities, the identity cos

2

-

= sin(). In

order to see that this identity is true, we'll start with cos

2

-

and we'll

use that cosine is an even function, so

cos - = cos - - = cos -

2

2

2

Now we can use the identity cos

-

2

= sin() (which is Lemma 9 from

the Sine and Cosine chapter) so that we have

cos - = cos - = sin()

2

2

as we had claimed.

Using the cofunction identity that we just examined, sin() = cos

2

-

,

we can check that the first cofunction identity from the list above is true:

sin - = cos - - = cos()

2

22

276

Exercises

For #1-12, use the chart on page 227 in the chapter "Sine and Cosine" and

that

csc()

=

1 sin()

,

sec()

=

1 cos()

,

and

cot()

=

cos() sin()

to

find

the

given

value.

1.) csc

6

2.) csc

4

3.) csc

3

4.) csc

2

5.) sec(0)

6.) sec

6

7.) sec

4

8.) sec

3

9.) cot

6

10.) cot

4

11.) cot

3

12.) cot

2

Find the solutions of the following equations in one variable. 13.) loge(x) = loge(12) - loge(x + 1) 14.) (x - 4)2 = 36 15.) e3x-2 = -4

277

Match the numbered piecewise defined functions with their lettered graphs below.

16.) f (x) =

csc(x) 2

if

0

<

x

<

2

or

2

<

x

<

if

x

=

2

17.) g(x) =

csc(x) -2

if

0

<

x

<

2

or

2

<

x

<

if

x

=

2

18.) h(x) =

I

cot(x) 2

if

0

<

x

<

2

or

2

<

x

<

if

x

=

2

19.) p(x) =

cot(x) -2

if

0

<

x

<

2

or

2

<

x

<

if

x

=

2

I

A.)

B.)

I

I

C.)

D.)

I

I

I

I

278

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