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Chapter Eight
Trigonometric Functions
The trigonometric functions sine and cosine are also called circular functions
since they are defined in terms of distances around a unit circle, a circle with
a radius equal to one. Distances are measured around this circle in a
counterclockwise direction, starting at the initial point (1, 0). Since the
length of the circumference of a circle is the product of
and the radius, the
distance around the unit circle is
circle is a distance of
. One-quarter of the way around the
, one-half the way around is
, and one and a half
times around the circle is . If you travel in a clockwise direction around the
circle, the distances are negative numbers. The distance from (1, 0) to (0, –1)
in the clockwise direction is
.
How far around the circle do you
travel from (0, 1) to (–1, 0):
a. in the positive direction
b. in the negative direction
c. in the positive direction,
passing through (1, 0) twice.
Ans.:
Measurement around the circle is called radian measure. A radian is equal
to the length of the radius. For the unit circle, one radian is equal to one.
You can mark off a radian around a circle by taking one of the radius lines and
wrapping it around the circle. The circumference for any circle is
radians, which is
for the unit circle. Be sure that you have your
calculator in radian mode instead of degree mode before you perform any
calculations with the circular functions.
,
,
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
The circular functions sine and cosine give the coordinates of the point on the
unit circle you reach when you travel around the unit circle a given distance.
If the distance you go around the circle from the initial point (1, 0) is x, then
the sine (written sin x) is the second coordinate of the point that you reach and
the cosine (written cos x) is the first coordinate of that point.
You can see several elementary properties of these functions in the unit circle
representation above.
Properties of Sine and Cosine
First, since sin x and cos x are defined for values of x that represent distances
traveled on the unit circle, you can see that neither can ever take on a value
less than –1 nor greater than 1. No points on the unit circle could ever have
coordinates outside these numbers.
–1
sin x
1
–1
cos x
1
Start at the initial point (1, 0) and move in the positive (counterclockwise)
direction. The values of sin x (the second coordinate of the point on the
circle) increase from 0 to 1 as x (the distance along the circle) increases from
0 to
to
. Then the sine values decrease from 1 to –1 when x changes from
, and increase from –1 back to 0 as x goes from
to
.
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
137
That takes you back to the point on the circle where you started, and the sine
values repeat through the same numbers as x changes from
to
, since
this takes you around the circle again. Every time you change the value of x
by exactly , the value of sin x remains unchanged. The sine function is a
cyclic function with a period of
, which means that it takes on the same
sequence of values in every interval of x-values that is
long.
The cosine values you obtain as you travel in the positive direction around the
circle start at 1 at the initial point, decrease to –1 as x varies from 0 to , then
increase from –1 to 1 as x changes from
to
. The cosine values also
repeat after every complete trip around the circle, so the cosine function is also
cyclic with a period of
.
For any integer n:
sin (x + n @
) = sin x
cos (x + n @
) = cos x.
You can easily read off some values of sine and cosine from the unit circle.
Find: cos
, sin
sin 0 = 0, cos 0 = 1
sin
= 1, cos
=0
sin
= 0, cos
= –1
= –1, cos
=0
sin
Ans.: 0, 1, –1.
Because of the cyclic properties of these functions, you can also find the sine
and cosine for any distances around the circle, in either direction, which are
integer multiples of
sin
or
:
= –1, cos
= 1, sin
= 0.
You can see that the values of cosine are the same for a given distance around
the circle, regardless of whether the direction taken is positive or negative.
Since the first coordinate of the point on the circle gives the cosine, the value
is the same moving clockwise or counterclockwise the same distance. The
cosine function is an even function.
However, the second coordinates are different for clockwise and
counterclockwise paths. The values of the sine function take different signs
when you take different directions around the circle. The sine function is an
odd function.
cos (–x) = cos x and sin (–x) = –sin x
, cos
.
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Moving along the unit circle, you can see another pattern in the values of the
circular functions. The values obtained by the sine function are the same as
those obtained by the cosine function at the point
circle. For example, sin
sin
= –1 = cos
= 1 = cos 0, sin
, and sin
= 0 = cos
units earlier along the
= 0 = cos
,
. Looking at the unit
circle, you can see that this is true everywhere around the circle.
sin x = cos (x –
or
cos x = sin (x +
),
Note that this means the graph of
sin x is a translation of the graph of
)
cos x to the right
units, or the
graph of cos x is a translation of the
graph of sin x to the left
If you divide the unit circle into fourths, each quarter is called a quadrant of
the circle. The first quadrant consists of the points that are between 0 and
from the initial point, the second of the points between and , etc. You
can see from the unit circle diagram that both sine and cosine are positive in
the first quadrant, the sine is positive and the cosine negative in the second,
both are negative in the third, and the sine negative and cosine positive in the
fourth quadrant.
Fitting a right triangle into the first quadrant,
and using the Pythagorean Theorem, you can establish the most important
trigonometric identity,
, for all values of the variable x.
units.
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
When working with triangles, you measure the angles in degrees instead of the
radians you use to measure distances around circles. A right angle contains
90°, a straight angle contains 180°, and a circular angle contains 360°.
Therefore, when you travel completely around a circle, thedistance you travel
is
radians, and the size of the angle you have passed through is 360°.
Since a distance of
1 radian =
or
Change to radians:
45°, 60°, 150°
Ans.:
radians is equivalent to an angle of 360°,
degrees
,
1 degree =
or
,
.
Change to degrees:
,
and
139
,
radians.
Among the right triangles that can be fit into the unit circle are a few that you
have previously studied. A right triangle having two equal sides has two 45°
angles. A 45° angle is equivalent to a distance of
radians, or
Ans.: 30°, 72°, 315°.
th of the
way around a circle. Since each of the two sides of the triangle has the same
length, say l, the Pythagorean Theorem says that the hypotenuse has a length
of
=
l. The hypotenuse has a length equal to
times the length
of each other side. In order for such a right triangle to fit into the unit circle,
the hypotenuse (the radius of the circle) must be 1. Therefore, each side must
have length equal to
=
l
.
l
The coordinates of the point
radians along the unit circle are
,
Since the first coordinate gives the cosine and the second the sine at x =
cos
= sin
=
.
.
,
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Similarly, a right triangle that contains 30° and 60° angles has sides that are in
the ratio of
1:
: 2 (side opposite 30° angle : side opposite 60° angle: hypotenuse).
In order for such a triangle to fit into the unit circle, the hypotenuse must have
length 1, so that the side opposite the 30° angle must be
opposite the 60° angle must be
and the side
.
If the triangle is fit into the unit circle so that the 30° angle is placed at the
center of the circle, the distance around the circle from the initial point to the
end of the hypotenuse is
or
The coordinates of the point
giving cos
=
, and sin
th of the circle, or
radians.
radians along the unit circle are
=
,
,
.
On the other hand, if the triangle is fit into the unit circle so that the 60° angle
is placed at the center of the circle, the distance around the circle from the
initial point to the end of the hypotenuse is
radians.
or
th of the circle, that is,
2
1
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
The coordinates of the point
you have cos
=
, and sin
radians along the unit circle are
=
,
141
and
.
To summarize, you should now be familiar with the cosines and sines of the
following x-values:
You should make an effort to
remember the information in this
table. The values of the sines and
cosines for these x-values and their
integer multiples are the only ones
that you need to know for your
trigonometric work.
By using the unit circle, you can also determine the cosine and sine of any
integer multiple of these x-values.
For example, to find sin
, you travel in a clockwise direction around
the unit circle once (a distance of
, or
), then
more in the same
clockwise direction. Since you end at the same position you reach
traveling
in the positive direction, sin
= sin
= –1.
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
To find cos
, you move around the circle into the second quadrant.
After traveling a distance of
around the circle you find yourself at a
point with a negative first coordinate. Since you have gone
past x = ,
and the unit circle is a symmetric figure, you are now as far to the left of
the center of the circle as you would be to the right of the center for x =
. From this you can see that
cos
Find sin
= –cos
=
.
. You want the second coordinate at the point you reach
after traveling a distance
around the circle in the positive direction.
This puts you in the third quadrant, where the sine is negative. You are
now as far below 0 as the point
of the way around is above 0.
Therefore,
sin
Find cos
= –sin
=
.
. You must now travel in the counterclockwise direction
a distance of
additional
. Go once completely around the circle (
) and an
past where you started. The first coordinate at this point is
the same as the first coordinate of the point that is
from the initial point
in the positive direction.
cos
= cos
=
.
You can find the value of the trigonometric functions for any integer
multiples of
,
,
, and
Find: sin
, sin
in the same way. Move around the circle the
distance you wish and see which quadrant you stop in. Let the circle show
you which sign is the proper one to use for your value.
Ans.:
,
,
, cos
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Graphing Sine and Cosine
Using the information from the first part of this chapter and the unit circle, you
can obtain graphs for the sine and cosine functions.
Both sine and cosine are defined for all x-values, since you can travel any
distance you wish in either direction around the circle, and the point you reach
has both a first and a second coordinate defined.
To graph f1(x) = sin x, you may first find the intercept point. Earlier you
saw in the unit circle that sin 0 = 0.
You cannot say much about end-behavior, because the sine function is
cyclic. Moving through larger x-values takes you around the circle again,
either in the positive or negative direction. However, all values for the
sine function are between –1 and 1.
You have already seen that the sine function is an odd function, and that
it is cyclic with a period of
. Therefore, when you get the graph for xvalues between x = 0 and x =
entire graph.
, you know enough to determine the
Remember that as your distance around the circle in the positive direction
increases from 0 to
, sin x increases from 0 to 1. At first this increase
is rapid, because the circle rises steeply near your starting point. As you
get closer to x =
when x =
, the sine increases at a slower rate until it reaches 1
.
For distances around the circle greater than
, the sine values decrease
at the same rate that they previously increased, until the sine returns to 0
when x = .
The sine values then continue to decrease until x =
, where you reach
the lowest sine value, –1.
From this point, the sine values again increase, returning to 0 when
x=
.
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Once you have the graph from x = 0 to x = , you can either analyze the
change in sine values as you move in the negative direction around the
circle, or use the fact that the sine function is an odd function to obtain the
graph for negative x-values from x = 0 to x =
.
Since the sine function is cyclic with a period of
, you can obtain the
graph for the entire domain by repeating what you already have for every
additional distance of
you move in either direction.
Complete graph of f1(x) = sin x.
Graphs having this regular, symmetric, cyclic pattern are called sinusoidal
curves. Oscilloscopes often show such patterns, for example.
Obtaining the graph of f2(x) = cos x requires a similar analysis.
You can find the intercept by looking at the unit circle, cos 0 = 1.
As you move away from your initial point around the circle in the positive
direction, the first coordinate of your point decreases from 1, slowly at
first, then more rapidly as you get near x =
, where the cosine is 0.
The cosine values continue to decrease rapidly, then more slowly as you
get near x = , where the value of the cosine is –1.
From x =
to x =
, the cosine values increase at the same rate they
decrease between 0 and . At the end of one complete circuit around the
unit circle the cosine is again equal to 1.
This gives you the graph of one period of f2(x) = cos x.
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
145
To obtain the graph in the negative direction, you can use the fact that the
cosine is an even function and produce the other side of the graph using
the symmetry, or you can analyze the change in the cosine values as you
move around the circle in the counterclockwise direction. The cosines
decrease from 1 down to –1, which the function reaches when x =
,
then increase up to 1 again at x =
. Using more negative x-values
than this starts you into the next cycle.
You obtain the complete graph by repeating this graph through its next cycles
both for positive and negative x-values. Again, end-behavior is irrelevant
because the cosine function is cyclic.
Complete graph of f2(x) = cos x.
Since sin (x +
) = cos x, you can also obtain the graph of the cosine function
by shifting the graph of the sine function
units to the left.
Graph of cos x superimposed on the graph of sin x.
This property was established on page
138.
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
You can also obtain the graphs of trigonometric functions modified in other
ways besides linear translations.
Sketch the graphs of
a. –5 cos x
b.
sin x.
Multiplying the sine or cosine function by a number changes the range of
possible values [–1, 1] to multiples of these values. Such a multiple is called
the amplitude of the function.
For example, the graph of f3(x) = 4 cos x looks the same as the graph of
cos x, but each value is 4 times greater, so that the maximum and
minimum values for f3(x) are 4 and –4.
(a)
(b)
Graph of 4 cos x compared to cos x.
Multiplying the variable by a number before taking the sine or cosine changes
the speed you travel around the circle, and therefore changes the period of
each cycle of the function.
Sketch the graphs of:
a. cos
x
b. 3 sin 4x
For example, let f4(x) = sin 2x. Since each x-value is doubled, you are
twice as far along the circle as you would be in sin x for the same x-value.
You return to your initial point on the circle in one-half the time. As x
changes from 0 to
, 2x changes from 0 to
. The period of this
function is rather than . Otherwise, the graph looks the same as sin
x with maximum value 1 and minimum value –1.
(a)
Graph of sin 2x compared to sin x.
(b)
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
When you add or subtract a number from the trigonometric function, the
entire graph rises or falls that number of units.
Sketch the graph of
+ sin x.
For example, let f5(x) = 2 sin x – 3. There are several steps to obtaining
the graph of this function from the graph of sin x. The coefficient of x, ,
slows down your trip around the unit circle to half the usual speed. This
makes the period twice as long as usual, since x must go from 0 to
in
order for x to travel from 0 to
. The coefficient of sin x, 2, doubles
the amplitude. The graph cycles between a low of –2 and a high of 2.
Subtracting 3 from each trigonometric value moves the entire graph
downward 3 units.
Sketch the graphs of
a. 2 cos (x – 4)
b. sin (x +
Graph of f5(x) compared to sin x.
Another example that combines several of these elementary
transformations is f6(x) = 1 –
cos 3(x –
). Compared to the graph of
cos x, f6(x) is reflected across the x-axis, has the amplitude reduced to ,
has the period changed to
often, moves
which makes the graph cycle 3 times as
units to the right and 1 up.
(a)
(b)
Graph of f6(x) compared to cos x.
)
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Other Trigonometric Functions
The other trigonometric functions, tangent, cotangent, secant and cosecant, are
defined in terms of the sine and cosine functions. You can obtain the
properties and graphs of these functions by using your knowledge of the sine
and cosine.
Your variable still represents the distance traveled in the counterclockwise
direction around a unit circle. The tangent (written tan x), cotangent (cot x),
secant (sec x) and cosecant (csc x) are all defined as rational functions of the
sine and cosine.
tan x =
, cot x =
sec x =
, csc x =
Note that cot x =
csc x =
, not
cos x =
, not
Placing a right triangle inside the unit circle,
you can see that the tangent
is the ratio of the two sides of the triangle
that have lengths equal to sin x and cos x. These are the side opposite the
angle placed at the center of the circle and the side adjacent to this angle.
Similarly, the cotangent is the ratio of the length of the adjacent side to the
length of the opposite side, the secant is 1 (the length of the hypotenuse of the
triangle) divided by the length of the adjacent side, and the cosecant is the
length of the hypotenuse divided by the length of the opposite side. These
triangle relationships hold even when the triangles are not bounded by the unit
circle.
tan x =
cot x =
sec x =
csc x =
, but
and
.
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Graphs of the Other Trigonometric Functions
You can obtain the graphs for tan x, cot x, sec x and csc x by using the
properties you have already derived from the unit circle for sin x and cos x.
For the graph of tan x, note that you have a rational function, with cos x in the
denominator. The first thing you do to graph any rational function is to
eliminate any values of the input variable which would produce a value of
zero in the denominator. Looking at the unit circle, you can see that cos x is
zero each time you reach the end of the first and third quadrant, traveling in
the positive direction. The values of x at these points include ,
and all the other odd integer multiples of
The intercept, tan 0 =
=
,
,
. Traveling in the clockwise
direction, you reach the same points when x is
other negative odd integer multiples of
,
,
,
and all the
.
= 0. Notice that tan x is also 0 when x =
,
,
,
and any other even integer multiple of
because when you
travel these distances around the unit circle you return to the initial point
where the sine is 0 and the cosine is 1. You can also see that tan x =0 at the
odd multiples of , since the sine is also equal to 0 there, while the cosine is
–1. Since these are the only places where the sine is 0, they are the only values
for which the tangent is equal to 0. The graph of the tangent crosses the axis
when x is an integer multiple of and at no other values.
When you use values just before x =
, still in the first quadrant, both sine
and cosine are positive and so is the tangent. However, the cosine is close to
0, so the tangent is a large positive number for these values. On the other side
of x =
, the cosine is negative although the sine is still positive (near 1),
causing the tangent to be negative. The tangent is again large in size, since the
cosine is close to 0 if you have not gone far past x =
.
Continuing around the circle, the tangent stays negative through the second
quadrant, but becomes less negative (closer to 0) as the sine approaches 0 and
the cosine nears –1. As you have already seen, the tangent equals 0 at x = .
Continuing into the third quadrant, the tangent becomes positive, since both
sine and cosine are negative.
149
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
The tangent values are increasing since the sine heads toward –1 while the
cosine becomes closer to 0. Dividing by the small cosine values gives you
large positive quotients for the tangent values as the x-values get closer to
x=
, which is a value that is not in the domain of tan x.
On the other side of x =
, the tangent is negative, since the sine is negative
and the cosine positive. Near x =
, the tangent has very negative values,
since the cosine in the denominator is close to 0, but as the cosine increases
toward 1 and the sine approaches 0, the tangent also approaches a value of 0,
which it attains when x =
.
Now you are back to where you started, and the values for the tangent cycle
through the same sequence of values as x increases from
to
. You
started a new cycle of values at x =
, and you can see from the graph you
have drawn that the period of the tangent function is rather than the period
of both sine and cosine.
Going in the negative direction, you obtain the same tangent values, but in the
opposite order. The tangent is negative in the fourth quadrant, positive in the
third, then negative in the second and positive in the first. When you return
to the initial point after traveling a distance of
, you have completed two
cycles and are back where you started.
Complete graph of tan x.
Notice that the tangent can take on any positive or negative value, unlike sine
and cosine, which must have values between –1 and 1.
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Since sin (–x) = –sin x and cos (–x) = cos x,
tan (–x) =
=
= –tan x.
The tangent function is an odd function, as you can also see from the graph.
Cot x =
is also an odd function, and has a graph that you can obtain from
that of the tangent by noting that both tan x and cot x have the same signs in
each quadrant, but the cotangent is large when the tangent is near zero, and
vice-versa.
Analyzing cot x =
separately from tan x, you first note that any value of
x for which sin x = 0 is not in the domain of cot x. The points at which sin x
is zero include x = 0, ,
,
, and all other integer multiples of . There
is no intercept. The cotangent function has a value of zero when cos x is zero.
This occurs when x is any odd integer multiple of
. Note that this is exactly
the opposite of the situation for the tangent function, which is the reciprocal
of the cotangent.
As you move through the first quadrant, cos x decreases from 1 to 0 and sin x
increases from 0 to 1. Since the numerator of cot x is decreasing while the
denominator is increasing, the function values rapidly decrease to 0, which
you reach when x =
.
In the second quadrant, the cotangent becomes negative, and since the sin x
term in the denominator decreases, the size of these negative values becomes
greater as you approach x = , for which the function does not exist.
In the third quadrant, both sine and cosine are negative. The cosine heads
toward zero while the size of the sine increases toward –1. This combination
produces positive cotangent values that decrease to 0 at x =
.
When you pass into the fourth quadrant, the cotangent values become
increasingly negative again since the sine remains negative, but gets closer to
zero. This makes the cotangent values drop off rapidly in the negative
direction. As at x= 0, cot x does not exist at x=
, but beyond this value it
cycles through the same values as on your preceding trip around the circle.
Moving in the clockwise direction produces the same results in the opposite
order.
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Complete graph of cot x.
Note from the graph that, like tan x, the period of cot x is
that cot x is another odd function.
To obtain the graph of sec x, use its definition sec x =
. You can also see
. You must
eliminate the same points from the domain of sec x as from tan x, since both
these functions have a denominator of cos x. This means that again you have
vertical asymptotes at all the odd multiples of
The intercept is
=
.
= 1. You also obtain this value whenever x is an
integer multiple of . Since all cosine values lie between –1 and 1, the secant
values are reciprocals of those numbers, namely numbers greater than or equal
to 1 and less than or equal to –1. Since cosine values are never greater than
1 nor less than –1, secant values are never between –1 and 1.
As you move around the first quadrant, the cosine values decrease from 1 to
0. Simultaneously, their reciprocal values increase from 1. Secant values are
steadily increasing from 1 until you approach x =
, for which there is no
secant value defined.
In the second quadrant, cosine (and therefore secant) values are negative. The
cosine values are near zero at first, causing the secant to be extremely
negative, then the cosine approaches –1, which it reaches at x = . The
secant meanwhile becomes less negative until it also reaches –1 when you get
to x = .
cot (–x) =
=
= –cot x
CHAPTER 8: TRIGONOMETRIC FUNCTIONS
153
The cosine remains negative in the third quadrant. It retraces the same values
(in the opposite order) that has in the second quadrant. So does the secant,
although the secant does not exist at x =
and becomes extremely negative
as you approach this x-value.
In the fourth quadrant, the cosine returns to positive values, increasing to 1 at
x=
. The secant is also positive, very large when x is just past
, and
decreasing to reach 1 when x =
. The secant also repeats its values again
as you make the next circuit of the circle. Proceeding in the counterclockwise
direction again produces all the same values in the opposite order.
Complete graph of sec x.
Another way to obtain the graph of the secant is to take advantage of the fact
that sec x is the reciprocal of cos x. Any time that a function reaches the value
of zero, its reciprocal does not exist. At a point eliminated from the domain
of the function because it causes a division by zero, the reciprocal value is
zero). When a function is equal to either of the values –1 or 1, so is the
reciprocal. The reciprocal function is positive or negative when the original
function is. Input values that produce large output values for the original
function are small (closer to zero) in the reciprocal and vice-versa.
Using these facts allows you to sketch the graph of a reciprocal function with
the graph of the original by intersecting them at function values of –1 and 1,
producing vertical asymptotes for one when the other is zero and large values
when the other is small, keeping their signs the same.
One example of such a graph is on the right and the graph of sec x drawn as
the reciprocal of cos x is shown on the following page.
Graph of a function and its reciprocal.
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CHAPTER 8: TRIGONOMETRIC FUNCTIONS
Sketch the graph of csc x.
Graph of csc x given with sin x.
Graph of sec x drawn as the reciprocal of cos x.
The graphs of tangents, cotangents, secants and cosecants change by
translations and stretches in the same ways that the sine and cosine graphs do.
The graph of tan
x looks like that of tan x, but the coefficient of x,
,
decreases your speed moving around the unit circle. Instead of a period of
, you have a period of
. The graph does not complete a full cycle until
the x-values have gone through an interval of width
Graph of tan .25x.
.
Graph of tan x drawn with the same scale.