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Trig functions and x and y
In this presentation we describe the graphs of each of the six trig
functions. We have already focused on the sine and cosine functions,
devoting an entire lecture to the sine wave. Now we look at the tangent
function and then the reciprocals of sine, cosine and tangent, that is,
cosecant, secant and cotangent.
Elementary Functions
Part 4, Trigonometry
Lecture 4.5a, Graphing Trig Functions
First a note about notation. Up to this time we have viewed trig functions
as functions of an angle θ and have tended to reserve the letters x, y for
coordinates on the unit circle. But 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 θ and y is
the old xy !)
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU)
Elementary Functions
2013
1 / 22
The tangent function
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Elementary Functions
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The tangent function
sin x
The tangent function tan x = cos
x has a zero wherever sin x = 0, that is,
whenever x is . . . , −2π, − π, 0, π, . . . , πk, ... (where k is an integer.)
Here is the graph of the tangent function.
The tangent function is undefined whenever cos x = 0, that is, at the
(2k+1)π
π π 3π 5π
x-values . . . , − 3π
, ... (where k is an integer.)
2 , − 2, 2, 2 , 2 ,...,
2
Indeed, at these x-values, the tangent function has vertical asymptotes.
Smith (SHSU)
Elementary Functions
2013
3 / 22
Smith (SHSU)
Elementary Functions
The tangent function
Central Angles and Arcs
When we discussed the sine wave, we also discussed concepts of period,
amplitude and phase shift.
The domain of the tangent function is all real numbers except those where
cos x = 0.
The graph of y = sin x has period 2π, amplitude 1 and phase shift 0.
We can write this in set notation as
We observed earlier that the tangent function has period π. 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, (−∞, ∞).
. . . (−
π π
π 3π
3π 5π
3π π
, − ) ∪ (− , ) ∪ ( , ) ∪ ( , ) . . . .
2 2
2 2
2 2
2
2
Since this domain is a union of an infinite number of open intervals (each
interval of length π) then we might write this union in a more compact
form using a more general “iterated union” notation:
Domain of the tangent function =
∞
[
k=−∞
(
(2k − 1)
(2k + 1)
π,
π).
2
2
(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.)
Smith (SHSU)
Elementary Functions
2013
5 / 22
Smith (SHSU)
Elementary Functions
2013
The graphs of secant and cosecant
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.
Here is the graph of the secant function (in blue) with asymptotes as
dotted red lines and the cosine function hiding in light yellow.
Smith (SHSU)
Elementary Functions
2013
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Similarly whenever the cosine is negative (but not less than −1) the secant
function is negative but less than or equal to −1.
7 / 22
Smith (SHSU)
Elementary Functions
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8 / 22
The graphs of secant and cosecant
The graphs of secant and cosecant
The graph of the cosecant function is similar to the graph of the secant
function. The cosecant function is the reciprocal of the sine function.
When we investigated the sine and cosine functions we observed that the
cosine function is the sine function shifted to the left by π2 (that is,
cos x = sin(x + π2 )) and so the graph of the sine function is the same as
the graph of the cosine function shifted to the right by π2 .
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 .
Smith (SHSU)
Elementary Functions
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Smith (SHSU)
Elementary Functions
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The cotangent function
The cotangent function
The cotangent is the reciprocal of tangent.
Here is the graph of the cotangent function.
The cotangent is the reciprocal of tangent. 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 π2 and then
reflecting it across the x-axis.
π
cot(x) = − tan(x + ).
2
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Elementary Functions
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Smith (SHSU)
Elementary Functions
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Tangent and cotangent
Tangent and cotangent
Another way to look at the cotangent function: since cos x = sin(x + π2 )
and that − sin x = sin(x + π) then
sin(x + π2 )
sin(x + π2 )
sin(x + π2 )
cos x
π
cot(x) =
=
=−
=−
π = − tan(x+ ).
sin x
sin x
sin(x + π)
cos(x + 2 )
2
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Elementary Functions
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Smith (SHSU)
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Graphs of the six trig functions
Elementary Functions
In the next presentation, we work through some exercises with the graphs
of the six trig functions.
Part 4, Trigonometry
Lecture 4.5b, Graphing Trig Functions: Some Worked Problems
(End)
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU)
Elementary Functions
2013
15 / 22
Smith (SHSU)
Elementary Functions
2013
16 / 22
Some worked problems
Some worked problems
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.
1
y = tan(x − π2 )
2
y = tan(2x − π2 )
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.
3
y = 5 tan(x − π2 ) + 1
4
y = −2 tan(2x − π2 ) + 4
Solutions.
Solutions.
1
2
3
To graph y = tan(x − π2 ), shift the graph of the tangent function
right by π2 .
To graph y = tan(2x − π2 ) = tan(2(x − π4 )), shift the graph of the
tangent function right by π4 and then shrink the function by a factor
of two in the horizontal direction (centered about the line x = π4 .)
Smith (SHSU)
Elementary Functions
2013
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Some worked problems
To graph y = −2 tan(2x − π2 ) + 4 = −2 tan(2(x − π4 )) + 4, shift the
graph of the tangent function right by π4 , then shrink the function by
a factor of two in the horizontal direction, stretch it by a factor of 2
in the vertical direction, reflect it across the x-axis, and then shift it
up by 4.
Smith (SHSU)
Elementary Functions
2013
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Two more worked problems
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.
5 y = cot(x)
π
6 y = cot(x − 2 )
π
7 y = cot(2x − 2 )
Solutions.
π
5 Since cot(x) = − tan(x + 2 ) then to graph y = cot(x), reflect the
graph of y = tan x across the x-axis and shift it left by π2 .
π
6 To graph y = cot(x − 2 ), first reflect the graph of y = tan x 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 .
π
7 To graph y = cot(2x − 2 ), first reflect the graph of y = tan x 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)
4
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.
Elementary Functions
2013
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8
Find all solutions to the trig equation tan θ = 1
Solution. From looking at the unit circle, we see that θ = 45◦ = π/4
is a solution to this equation. So also is θ = 225◦ = 5π/4, the angle
in the third quadrant with reference angle π/4. But there are many
more solutions; if we add 2π to θ, we get new angles that satisfy this
equation. Therefore
{
π
5π
+ 2πk : k ∈ Z} ∪ {
+ 2πk : k ∈ Z}
4
4
is the (infinite) set of all solutions.
However, recall that the tangent function has period π. So we could
simplify this answer by just writing
{
Smith (SHSU)
π
+ πk : k ∈ Z}
4
Elementary Functions
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Two more worked problems
9
Worked problems on graphing trig functions
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) = 21 . Since the
tangent of θ is negative then θ is in the fourth
√
π
3
assume θ = −30◦ = − . Then sin(θ) = −
and tan(θ) = − 3
3
2
and the other functions are reciprocals of these.
Smith (SHSU)
Elementary Functions
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In the next presentation, we will look at inverse trig functions, that is, the
inverse functions of cosine, sin, tangent, secant, cosecant and cotangent.
(End)
Smith (SHSU)
Elementary Functions
2013
22 / 22
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