4.5 Graphing Trig Functions - Sam Houston State University

Elementary FunctionsPart 4, Trigonometry

Lecture 4.5a, Graphing Trig Functions

Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 1 / 22

Trig functions and x and y

In this presentation we describe the graphs of each of the six trigfunctions. We have already focused on the sine and cosine functions,devoting an entire lecture to the sine wave. Now we look at the tangentfunction and then the reciprocals of sine, cosine and tangent, that is,cosecant, secant and cotangent.

First a note about notation. Up to this time we have viewed trig functionsas functions of an angle θ and have tended to reserve the letters x, y forcoordinates on the unit circle. But it is time to return to our originalcustom about variables in functions, using x as the input variable and y asthe output variable.For example, when we write y = tan(x) we now think of x as an angle andy as a ratio of two sides of a triangle. (In this case x is the old θ and y isthe old y

x !)


The tangent function

The tangent function tanx = sinxcosx has a zero wherever sinx = 0, that is,

whenever x is . . . ,−2π, − π, 0, π, . . . , πk, ... (where k is an integer.)

The tangent function is undefined whenever cosx = 0, that is, at thex-values . . . ,−3π

2 , −π2 ,

π2 ,

3π2 ,

5π2 , . . . ,

(2k+1)π2 , ... (where k is an integer.)

Indeed, at these x-values, the tangent function has vertical asymptotes.



Here is the graph of the tangent function.



When we discussed the sine wave, we also discussed concepts of period,amplitude and phase shift.

The graph of y = sinx has period 2π, amplitude 1 and phase shift 0.

We observed earlier that the tangent function has period π. This is clearfrom the unit circle definition of tangent and this period is visible in ourgraph.It does not make sense to discuss the amplitude of the tangent functionsince the range of tangent is the full set of all real numbers, (−∞,∞).


Central Angles and Arcs

The domain of the tangent function is all real numbers except those wherecosx = 0.

We can write this in set notation as

. . . (−3π

2,−π

2) ∪ (−π

2,π

2) ∪ (

π

2,3π

2) ∪ (

3π

2,5π

2) . . . .

Since this domain is a union of an infinite number of open intervals (eachinterval of length π) then we might write this union in a more compactform using a more general “iterated union” notation:

Domain of the tangent function =

∞⋃k=−∞

((2k − 1)

2π,

(2k + 1)

2π).

(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.)


The graphs of secant and cosecant

The secant function is the reciprocal of cosine and so it has verticalasymptotes wherever cosx = 0.

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



Since −1 ≤ cosx ≤ 1 then the reciprocal function, secant, is boundedaway from the x-axis; whenever cosx 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 secantfunction is negative but less than or equal to −1.



The graph of the cosecant function is similar to the graph of the secantfunction. The cosecant function is the reciprocal of the sine function.



When we investigated the sine and cosine functions we observed that thecosine function is the sine function shifted to the left by π

2 (that is,cosx = sin(x+ π

2 )) and so the graph of the sine function is the same asthe graph of the cosine function shifted to the right by π

2 .

If the graph of sine is achieved by shifting cosine to the right by π2 then

the graph of cosecant is the secant function shifted to the right by π2 .


The cotangent function

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


The cotangent function

The cotangent is the reciprocal of tangent. We see from looking at thegraph of cotangent that the graph of cotangent can be achieved by takingthe graph of the tangent function, moving it left (or right) by π

2 and thenreflecting it across the x-axis.

cot(x) = − tan(x+π

2).


Tangent and cotangent

Another way to look at the cotangent function: since cosx = sin(x+ π2 )

and that − sinx = sin(x+ π) then

cot(x) =cosx

sinx=

sin(x+ π2 )

sinx= −

sin(x+ π2 )

sin(x+ π)= −

sin(x+ π2 )

cos(x+ π2 )

= − tan(x+π

2).


Tangent and cotangent


Graphs of the six trig functions

In the next presentation, we work through some exercises with the graphsof the six trig functions.

(End)


Elementary FunctionsPart 4, Trigonometry

Lecture 4.5b, Graphing Trig Functions: Some Worked Problems

Dr. Ken W. Smith

Sam Houston State University

2013


Some worked problems

For each of the following functions, describe the transformation requiredto change the graph of the tangent function into the graph of theindicated function.

1 y = tan(x− π2 )

2 y = tan(2x− π2 )

Solutions.

1 To graph y = tan(x− π2 ), shift the graph of the tangent function

right by π2 .

2 To graph y = tan(2x− π2 ) = tan(2(x− π

4 )), shift the graph of thetangent function right by π

4 and then shrink the function by a factorof two in the horizontal direction (centered about the line x = π

4 .)




3 y = 5 tan(x− π2 ) + 1

4 y = −2 tan(2x− π2 ) + 4

Solutions.

3 To graph y = 5 tan(x− π2 ) + 1, shift the graph of the tangent

function right by π2 , stretch it vertically by a factor of 5 and then

move the function up 1.

4 To graph y = −2 tan(2x− π2 ) + 4 = −2 tan(2(x− π

4 )) + 4, shift thegraph of the tangent function right by π

4 , then shrink the function bya factor of two in the horizontal direction, stretch it by a factor of 2in the vertical direction, reflect it across the x-axis, and then shift itup by 4.




5 y = cot(x)

6 y = cot(x− π2 )

7 y = cot(2x− π2 )

Solutions.

5 Since cot(x) = − tan(x+ π2 ) then to graph y = cot(x), reflect the

graph of y = tanx across the x-axis and shift it left by π2 .

6 To graph y = cot(x− π2 ), first reflect the graph of y = tanx across

the x-axis and shift it left by π2 to obtain the graph of the cotangent

function. Finally, shift the graph right by π2 .

7 To graph y = cot(2x− π2 ), first reflect the graph of y = tanx across

the x-axis and shift it left by π2 to obtain the graph of the cotangent

function. Then shift the graph right by π4 and then shrink the

function by a factor of two in the horizontal direction.Smith (SHSU) Elementary Functions 2013 19 / 22

Two more worked problems

8 Find all solutions to the trig equation tan θ = 1Solution. From looking at the unit circle, we see that θ = 45◦ = π/4is a solution to this equation. So also is θ = 225◦ = 5π/4, the anglein the third quadrant with reference angle π/4. But there are manymore solutions; if we add 2π to θ, we get new angles that satisfy thisequation. Therefore

{π4+ 2πk : k ∈ Z} ∪ {5π

4+ 2πk : k ∈ Z}

is the (infinite) set of all solutions.

However, recall that the tangent function has period π. So we couldsimplify this answer by just writing

{π4+ πk : k ∈ Z}


Two more worked problems

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

Solution. Since the secant of θ is 2 then cos(θ) = 12 . Since the

tangent of θ is negative then θ is in the fourth quadrant and we may

assume θ = −30◦ = −π3. Then sin(θ) = −

√3

2and tan(θ) = −

√3

and the other functions are reciprocals of these.


Worked problems on graphing trig functions

In the next presentation, we will look at inverse trig functions, that is, theinverse functions of cosine, sin, tangent, secant, cosecant and cotangent.

(End)


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4.5 Graphing Trig Functions - Sam Houston State University