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Graphs of the trigonometric functions

Zeros of a function

The period of a function

The graph of y = sin x

The graph of y = cos x

The graph of y = sin ax

The graph of y = tan x

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

0,  ±π,  ±2π,  ±3π, .  .  .

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?

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,

(1) .  .  .  .  .  .  .  .   . sin θ = 0  when  θ = nπ

They will be the x-intercepts of the graph of the sine.

Line (1) will be true, moreover, for any argument θ.  For example,

sin 2x = 0  when 2x = nπ;

that is, when

Problem 1.   Where are the zeros of  y =sin 3x?

At 3x = nπ; that is, at

Which numbers are these?

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.

sin (θ + 2π) = sin θ.

sin θ therefore 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 graph of y = sin x

The zeros of y = sin x are at the multiples of π.  It is there that the graph crosses the x-axis, because there y = 0.  And its period is 2π. But what is the maximum value of the graph and what is its minimum value?

sin x has a maximum value of 1 at , and a minimum of −1 at —and at all angles coterminal with them.

Here is the graph of y = sin x:

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 15:  The unit circle.)

Problem 2. Vocabulary.

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

− < x < .

b)  What is the range of y = sin x?

−1 ≤ sin x ≤ 1.

The graph of y = cos x

The graph of y = cos x is the graph of y = sin x  translated units to the left.  It is sin (x + ).  And sin (x + )  =  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 3.

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

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

 Compare the graphs above.

Problem 4.   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?

c)  Where are its zeros?

Problem 5.   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?

c)  Where are its zeros?

Problem 6.   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?

The graph of y = tan x

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

Why is that the graph?  It has effectively been explained in the previous topic, where we considered the line value DE of tan x in quadrants IV and I.

In quadrant IV, tan x -- the line value DE -- is negative and takes on all possible negative values:

−∞ < tan x < 0.

At x = 0, tan x = 0.

And finally in quadrant I, tan x takes on all positive values:

0 < tan x < ∞.

And so in the interval from − to , tan x takes on all its possible values. That interval constitutes a complete period of y = tan x.

Here again is the graph.

At the quadrantal angles  − and , tan x has no value. Therefore the lines x = − and x = are vertical asymptotes.

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?

Next Topic:  Inverse trigonometric functions

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