draw the graph of function f(x)= cos(x) in the interval [ 0,pie] .
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Answer:
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. We include 0 as even.
2nπ also signifies any multiple of 2π. Any complete revolution.
θ and θ + 2nπ are therefore coterminal.
sin θ, therefore, is equal to sin (θ + 2nπ).
sine theta
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 number.
The graph of y cos x
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.
The period of a function
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 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, pi-2 units to the left.
For, sin (x + pi-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.)