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19
Graphs of the trigonometric functions
Zeros
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, . . .
By '2n' we mean to signify an even number.
b) (2n + 1)π
The odd multiples of π:
±π, ±3π, ±5π, ±7π, . . .
By '2n + 1' we mean to signify 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 16 on the unit circle that the value of sin θ is equal to the ycoordinate. 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
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
Which numbers are these?
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 xaxis. 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.
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 xaxis.
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.
Definition. If, for all numbers 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,
For, sin (x + 
π 2 
) = cos x. The student familiar with the sum

formula can easily prove that. (Topic 20.)
Again, the height of the curve at every point is the line value of the cosine.
The graph of y = sin ax
When a function has this form,
y = sin ax,
then the constant a indicates the number of periods in an interval of length 2π.
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.
(In physics, when the independent variable is the time t, 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?
c) Where are its zeros?
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?
c) Where are its zeros?
Problem 7. y = sin ¼x.
a) What does ¼ indicate?
In an interval of length 2π, there 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
Consider the line value DE of tan x in the 4th and 1st quadrants:
As x goes from − 
π 2 
to 
π 2 
, DE  tan x  ranges through every real 
number.
− < tan x <
Quadrants IV and I comprise a complete period of the tangent function. Here is the graph:
Precalculus.)
Now in the 2nd Quadrant (Fig. 2), the graph has exactly the same negative values (KE') as in the 4th (DE).
And in the 3rd Quadrant (Fig. 3), the graph has exactly the same positive values (KE') as in the 1st (DE).
Thus the graph of Quadrants IV and I is repeated in Quadrants II and III, and periodically along the entire xaxis.
This is the graph of y = tan x.
Next Topic: Inverse trigonometric functions
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