Trigonometry

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18

Graphs of the trigonometric functions

Zeros of a function

The graph of y = sin x

The period of a function

The graph of y = cos x

The graph of y = sin ax

The graph of y = tan x


LET US BEGIN by introducing some algebraic language.  When we write "nπ," where n could be any integer, we mean "any multiple of π."

0,  ±π,  ±2π,  ±3π, .  .  .

Problem 1.   Which numbers are indicated by the following, where n could be any integer?

a)  2nπ

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The even multiples of π:

0, ±2π,  ±4π,  ±6π, .  .  .

2n, in algebra, typically signifies an even number.

b)  (2n + 1)π

The odd multiples of π:

±π,  ±3π,  ±5π,  ±7π, .  .  .

'

2n + 1 (or 2n − 1) typically signifies an odd number.

Zeros

By the zeros of sin θ, we mean those values of θ for which sin θ will equal 0.

Now, where are the zeros of sin θ?  That is,

sin θ = 0  when θ = ?

We saw in Topic 15 on the unit circle that the value of sin θ is equal to the y-coordinate.  Hence, sin θ = 0 at θ = 0 and θ = π -- and at all angles coterminal with them.  In other words,

sin θ = 0  when  θ = nπ.

This will be true, moreover, for any argument of the sine function.  For example,

sin 2x = 0  when the argument 2x = nπ;

that is, when

x  =  nπ
 2
.
Which numbers are these?  The multiples of  π
2
:
0,  ± π
2
,  ±π,  ± 3π
 2
, . . .

Problem 2.   Where are the zeros of  y = sin 3x?

At 3x = nπ; that is, at

x nπ
 3
.

Which numbers are these?

The multiples of  π
3
.

The graph of y = sin x

The zeros of y = sin x are at the multiples of π.  And it is there that the graph crosses the x-axis, because there sin x = 0.  But what is the maximum value of the graph, and what is its minimum value?

sin x has a maximum value of 1 at  π
2
, and a minimum value of −1
  at  3π
 2
 -- and at all angles coterminal with them.

Here is the graph of y = sin x:

The height of the curve at every point is the line value of the sine.

In the language of functions, y = sin x is an odd function. It is symmetrical with respect to the origin.
sin (−x) = −sin x.

y = cos x is an even function.

The independent variable x is the radian measure.  x may be any real

number.  We may imagine the unit circle rolled out, in both directions, along the x-axis.  (See Topic 14:  Arc Length.)

The period of a function

When the values of a function regularly repeat themselves, we say that the function is periodic.  The values of  sin θ  regularly repeat themselves

every 2π units.  Hence, sin θ is periodic.  Its period is 2π.  (See the previous topic, Line values.)

Definition.  If, for all values of x, the value of a function at x + p
is equal to the value at x --

If  f(x + p) = f(x)

-- then we say that the function is periodic and has period p.

The function  y = sin x  has period 2π, because

sin (x + 2π) = sin x.

The height of the graph at x is equal to the height at x + 2π -- for all x.

Problem 3.

a)  In the function y = sin x, what is its domain?

a)  (See Topic 3 of Precalculus.)

x may be any real number.  − < x < .

b)  What is the range of y = sin x?

sin x has a minimum value of −1, and a maximum of +1.

−1 y 1

The graph of y = cos x

The graph of y = cos x is the graph of y = sin x shifted, or translated,

   π
2
 units to the left.
For, sin (x π
2
)  =  cos x.  The student familiar with the sum

formula can easily prove that. (Topic 20.)

On the other hand, it is possible to see directly that

Topic 16.  Angle CBD is a right angle.

The graph of y = sin ax

Since the graph of  y = sin x  has period 2π, then the constant a in

y = sin ax

indicates the number of periods in an interval of length 2π.  (In y = sin x, a = 1.)

For example, if a = 2 --

y = sin 2x

-- that means there are 2 periods in an interval of length 2π.

If a = 3 --

y = sin 3x

-- there are 3 periods in that interval:

While if a = ½ --

y = sin ½x

-- there is only half a period in that interval:

The constant a thus signifies how frequently the function oscillates; so many radians per unit of x.

(When the independent variable is the time t, as it often is in physics, then the constant is written as ω ("omega"): sin ωt.  ω is called the angular frequency; so many radians per second.)

Problem 4.

a)   For which values of x are the zeros of y = sin mx?

At mx = nπ; that is, at x nπ
 m
.

b)   What is the period of y = sin mx?

   2π
 m
.  Since there are m periods in 2π, then one period is 2π

divided by m. Compare the graphs above.

Problem 5.   y = sin 2x.

a)   What does the 2 indicate?

In an interval of length 2π, there are 2 periods.

b)   What is the period of that function?

2π
 2
  = π

c)  Where are its zeros?

At x nπ
 2
.

Problem 6.   y = sin 6x.

a)   What does the 6 indicate?

In an interval of length 2π, there are 6 periods.

b)   What is the period of that function?

2π
  6
  =   π
3

c)  Where are its zeros?

At x nπ
 6
.

Problem 7.   y = sin ¼x.

a)   What does ¼ indicate?

In an interval of length 2π, there is one fourth of a period.

b)   What is the period of that function?

2π/¼ = 2π· 4 = 8π.

c)  Where are its zeros?

At x nπ
 ¼
  =  4nπ.

The graph of y = tan x

Here is one period of the graph of y = tan x:

Why is that the graph?  Consider the line value DE of tan x in the 4th and 1st quadrants:

As radian x goes from − π
2
 to  π
2
, tan x takes on all real values. That is,

for

π
2
  < x  <  π
2
,  

< tan x < .

Quadrants IV and I constitute a complete period of y = tan x.  In quadrant IV, tan x is negative; in quadrant I, it is positive; and tan 0 = 0. Again, here is the graph:

At the quadrantal angles  − π
2
 and  π
2
, tan x does not exist.  Therefore
  the lines  x = − π
2
  and  x π
2
 are vertical asymptotes. (Topic 18 of

Precalculus.)

Here is the complete graph of  y = tan x.

The graph of Quadrants IV and I is repeated in Quadrant II (where tan x is negative) and quadrant III (where tan x is positive), and periodically along the entire x-axis.

Problem 7.   What is the period of y = tan x?

One period is from − π
2
 to  π
2
. Hence the period is the

distance between those two points: π.


Next Topic:  Inverse trigonometric functions


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