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 = ?

The zeros of sine theta

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.

The zeros of sine theta

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?

maximum, minium values of sine x

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.

coterminal angles

Here is the graph of y = sin x:

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

The graph of y  cos x

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

The period of a function

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 period of a function

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.  −infinity < x < infinity.

b)  What is the range of y = sin x?

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

−1 less than or equal to y less than or equal to 1

The graph of y = cos x

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

sin (x + pi/2) = cos x

sin (x + pi/2) = cos x

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.

The period of y = sin 2x

If a = 3 --

y = sin 3x

-- there are 3 periods in that interval:

The period of y = sin 3x

While if a = ½ --

y = sin ½x

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

The period of y = sin 1/2 x

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:

The period of y = tan x

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

The period of y = tan x

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
,  

minus infinity < tan x < infinity.

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:

The period of y =  tan x

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 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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