4.5 Graphing All Six Trig Functions

4.5 Graphing All Six Trig Functions

In this section we take a little time to completely describe the graphs of the six trig functions. We have

already focused on the sine and cosine functions, devoting an entire lecture to the sine wave. So in this

section we will look at the tangent function and then the reciprocals of sine, cosine and tangent, that is,

cosecant, secant and cotangent.

But first a note about notation. Up to this time we have viewed trig functions as functions of an

angle and have tended to reserve the letters x, y for coordinates on the unit circle. Now, however, it is

time to return to our original custom about variables in functions, using x as the input variable and y as

the output variable.

For example, when we write y = tan(x) we now think of x as an angle and y as a ratio of two sides

of

a

triangle.

In

this

case

x

is

the

old

and

y

is

the

old

y x

!

4.5.1 The tangent function

The tangent function

tan x

=

sin x cos x

has

a zero wherever sin x

=

0, that is,

whenever

x is . . . , -2, -, 0, , . . . , k, ...

where k is an integer. The tangent function is undefined whenever cos x = 0, that is, at the x-values

.

.

.

,

-

3 2

,

-

2

,

2

,

3 2

,

5 2

,

.

.

.

,

(2k+1) 2

,

...

(where

k

is

an

integer.)

Indeed, at these x-values, the tangent

function has vertical asymptotes.

Here is the graph of the tangent function.

Figure 20. The tangent function (blue curves) along with vertical asymptotes (dotted red lines)

When we discussed the sine wave, we also discussed concepts of period, amplitude and phase shift. The graph of y = sin x had period 2, amplitude 1 and phase shift 0. We observed earlier that the tangent function has period . This is clear from the unit circle definition of tangent and this period is visible in our graph.

It does not make sense to discuss the amplitude of the tangent function since the range of tangent is the full set of all real numbers, (-, ).

The domain of the tangent function is all real numbers except those where cos x = 0. We can write this in set notation as

3

3 3 5

. . . (- , - ) (- , ) ( , ) ( , ) . . . .

22

22 2 2

22

Since this domain is a union of an infinite number of open intervals (each interval of length ) then we

172

might write this union in a more compact form using a more general "iterated union" notation:

(2k - 1) (2k + 1)

Domain of the tangent function =

(

,

).

2

2

k=-

(We won't do much with these more general arbitrary unions in this class, but it is important to see this notation once or twice in a precalculus class.)

4.5.2 The graphs of secant and cosecant

The secant function is the reciprocal of cosine and so it has vertical asymptotes wherever cos x = 0. Since -1 cos x 1 then the reciprocal function, secant, is bounded away from the x-axis; whenever cos x is positive (but no larger than 1) then the secant is positive but greater than or equal to 1. Similarly whenever the cosine is negative (but not less than -1) the secant function is negative but less than or equal to -1.

Here is the graph of the secant function (in blue) with asymptotes as dotted red lines and the cosine function hiding in light yellow.

Figure 21. The secant function with asymptotes

The cosecant function is similar. The cosecant function is the reciprocal of the sine function. When

we investigated the sine and cosine functions, earlier, we observed that the cosine function is the sine

function

shifted

to

the

left

by

2

(that

is,

cos x

=

sin(x +

2

))

and

so

the

graph

of

the

sine

function

is

the

same

as

the

graph

of

the

cosine

function

shifted

to

the

right

by

2

.

If

the

graph

of

sine

is

achieved

by

shifting

cosine

to

the

right

by

2

then

the

graph

of

cosecant

is

the

same

as

that

of

the

secant

function

shifted

to

the

right

by

2

.

The graph of the cosecant function, with asymptotes, is drawn in the figure below.

173

Figure 22. The cosecant function with asymptotes

4.5.3 The cotangent function The cotangent is the reciprocal of tangent. Here is the graph of the cotangent function.

Figure 23. The cotangent function

We see from looking at the graph of cotangent that the graph of cotangent can be achieved by taking the

graph

of

the

tangent

function,

moving

it

left

(or

right)

by

2

and

then

reflecting

it

across

the

x-axis.Thus

cot(x) = - tan(x + ).

2

Another

way

to

see

this

is

to

recognize

that

since

cos x

=

sin(x

+

2

)

and

that

- sin x

=

sin(x

+

)

then

cot(x)

=

cos x sin x

=

sin(x + sin x

2

)

=

sin(x -

+

2

)

sin(x + )

=

sin(x + -

cos(x +

2

)

2

)

=

- tan(x +

).

2

174

4.5.4 Some worked problems

1. For each of the following functions, describe the transformation required to change the graph of the tangent function into the graph of the indicated function.

(a)

y

= tan(x -

2

)

(b)

y

= tan(2x -

2

)

(c)

y

= 5 tan(x -

2

)

+

1

(d)

y

= -2 tan(2x -

2

)

+

4

(e) y = cot(x)

(f )

y

= cot(x -

2

)

(g)

y

= cot(2x -

2

)

Solutions.

(a)

To

graph

y

= tan(x -

2

),

shift

the

graph

of

the

tangent

function

right

by

2

.

(b)

To

graph

y

= tan(2x -

2

),

shift

the

graph

of

the

tangent

function

right

by

4

and

then

shrink

the function by a factor of two in the horizontal direction. (Or first shrink the function by a

factor

of

2

in

the

horizontal

direction

and

then

shift

it

right

by

2

.

(c)

To

graph

y

=

5 tan(x -

2

)

+

1,

shift

the

graph

of

the

tangent

function

right

by

2

,

stretch

it

vertically by a factor of 5 and then move the function up 1.

(d)

To

graph

y

=

-2 tan(2x -

2

)

+

4,

shift

the

graph

of

the

tangent

function

right

by

4

,

then

shrink the function by a factor of two in the horizontal direction, stretch it by a factor of 2 in

the vertical direction and then shift it up by 4.

(e)

To

graph

y

= cot(x),

reflect

the

graph

of

y

= tan x

across

the

x-axis

and

shift

it

left

by

2

to

obtain the graph of the cotangent function.

(f )

To

graph

y

= cot(x -

2

),

first

reflect

the

graph

of

y

= tan x

across

the

x-axis

and

shift

it

left

by

2

to

obtain

the

graph

of

the

cotangent

function.

Finally,

shift

the

graph

right

by

2

.

(g)

To

graph

y

=

cot(2x

-

2

),

first

reflect

the graph

of

y

= tan x

across

the

x-axis and shift

it

left

by

2

to

obtain

the

graph

of

the

cotangent

function.

Then

shift

the

graph

right

by

4

and

then

shrink the function by a factor of two in the horizontal direction.

2. Find all solutions to the trig equation tan = 1

Solution. From looking at the unit circle, we see that = 45 = /4 is a solution to this equation. So also is = 225 = 5/4, the angle in the third quadrant with reference angle /4. But there are many more solutions; if we add 2 to , we get new angles that satisfy this equation. Therefore

5

{ 4

+

2k

:

k

Z}

{

4

+ 2k : k Z}

is the (infinite) set of all solutions.

However, recall that the tangent function has period . So we could simplify this answer by just

writing

{ 4

+

k

:

k

Z}

175

3. The angle has the property that sec = 2 and tan is negative. Identify the angle and then find all six trig functions of the angle .

Solution.

Since

the

secant

of

is

2

then

cos()

=

1 2

.

Since

the

tangent

of

is

negative

then

is

in

3

the fourth quadrant and we may assume = - . Then sin() = - and tan() = 3 and the

3

2

other functions are reciprocals of these.

4.5.5 Other resources on the graphs of the six trig functions

In the free textbook, Precalculus, by Stitz and Zeager (version 3, July 2011, available at stitz-) this material is covered in section 10.5.

In the free textbook, Precalculus, An Investigation of Functions, by Lippman and Rassmussen (Edition 1.3, available at ) this material is covered in section 6.2.

In the textbook by Ratti & McWaters, Precalculus, A Unit Circle Approach, 2nd ed., c. 2014 (here at ) this material appears in section ?? of chapter 5. In the textbook by Stewart, Precalculus, Mathematics for Calculus, 6th ed., c. 2012 (here at ) this material appears in sections 5.3 and 5.4.

There are a collection of videos on graphing the various trig functions here at Khan Academy.

Homework. As class homework, please complete Worksheet 4.5, Graphs of the Six Trig Functions available

through the class webpage.

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