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Understanding Sine and Cosine: Period, Amplitude, and Transformations, Study Guides, Projects, Research of Trigonometry

An in-depth exploration of the sine and cosine functions, including their graphs, periods, amplitudes, and transformations. It covers the behavior of these functions from 0 to 2π, their x-intercepts, and the relationship between sine and cosine functions. The document also explains how to graph y = Asin(ωx + h) + v and y = Acos(ωx + h) + v, and discusses the significance of the frequency (ω) and period (T) in these equations.

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/12/2022

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2.6 Graphs of the Sine and Cosine Functions
x x y = sin x
0
0
SIN(B2)
If we graph y = sin x by
p
lottin
g
p
oints
,
we see
x x y = sin x
0
0
=
SIN(B2)
π/6 =PI()/6 =SIN(B3)
π/3 =PI()/3 =SIN(B4)
π/2 =PI()/2 =SIN(B5)
2π/3 =B5+PI()/6 =SIN(B6)
pgp,
the following:
Going from 0 to 2π,
sin(x) starts out with the
value 0, then rises to 1 at
π/2, then goes back to 0
A
i(θ
)
0 0 0.000
π/6 0.524 0.500
π/3 1.047 0.866
π/2 1.571 1.000
2π/3 2.094 0.866
5π/6 =B6+PI()/6 =SIN(B7)
π=B7+PI()/6 =SIN(B8)
7π/6 =B8+PI()/6 =SIN(B9)
4π/3 =B9+PI()/6 =SIN(B10)
3π/2 =B10+PI
()
/6 =SIN
(
B11
)
at π.
A
t
> π,
si
n
(θ
)
goes
from 0 to -1 at 3π/2, then
back to 0 at 2π. At this
point the sin values
repeat. The period of the
sine function is 2πTo
5π/6 2.618 0.500
π3.142 0.000
7π/6 3.665 -0.500
4π/3 4.189 -0.866
3π/2
4 712
-
1 000
()
()
5π/3 =B11+PI()/6 =SIN(B12)
11π/6 =B12+PI()/6 =SIN(B13)
2π=B13+PI()/6 =SIN(B14)
sine
function
is
2π
.
To
graph a more complete
graph of y= sin x, we
repeat this period in each
direction.
y = sin x
3π/2
4
.
712
1
.
000
5π/3 5.236 -0.866
11π/6 5.76 -0.500
2π6.283 0.000
π/3, 0.866
π/2, 1.000
2π/3, 0.866
1.000
1.500
0, 0.000
π/6, 0.500 5π/6, 0.500
π, 0.000 2π, 0.0000.000
0.500
7π/6, -0.500
4π/3, -0.866 5π/3, -0.866
11π/6, -0.500-0.500
3π/2, -1.000
-1.500
-1.000
pf3
pf4
pf5
pf8
pf9

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2.6 Graphs of the Sine and Cosine Functions

x x y = sin x

0 0 SIN(B2)

If we graph y = sin x by plotting points, we see

x x y = sin x 0 0 =SIN(B2)

π/6 =PI()/6 =SIN(B3)

π/3 =PI()/3 =SIN(B4)

π/2 =PI()/2 =SIN(B5)

2 π/3 =B5+PI()/6 =SIN(B6)

p g p , the following: Going from 0 to 2 π, sin(x ) starts out with the value 0, then rises to 1 at π/2, then goes back to 0 A i (θ )

0 0 0. π/6 0.524 0. π/3 1.047 0. π/2 1.571 1. 2 π/3 2.094 0. 5 π/6 =B6+PI()/6 =SIN(B7)

π =B7+PI()/6 =SIN(B8)

7 π/6 =B8+PI()/6 =SIN(B9)

4 π/3 =B9+PI()/6 =SIN(B10)

3 π/2 =B10+PI()/6 =SIN(B11)

at π. At x> π, sin(θ ) goes from 0 to -1 at 3π/2, then back to 0 at 2π. At this point the sin values repeat. The period of the sine function is 2π To

5 π/6 2.618 0. π 3.142 0. 7 π/6 3.665 -0. 4 π/3 4.189 -0. () ( ) 3 π/2 4 712 -1 000

5 π/3 =B11+PI()/6 =SIN(B12)

11 π/6 =B12+PI()/6 =SIN(B13)

2 π =B13+PI()/6 =SIN(B14)

sine function is 2π. To graph a more complete graph of y= sin x, we repeat this period in each direction. y = sin x

3 π/2 4 .712 1. 5 π/3 5.236 -0. 11 π/6 5.76 -0. 2 π 6.283 0.

π/3, 0.

π/2, 1. 2 π/3, 0.

0, 0.

π/6, 0.500 5 π/6, 0.

0.000 π, 0.000 2 π, 0.

7 π/6, -0.

4 π/3, -0.866 5 π/3, -0.

-0.500 11 π/6, -0.

3 π/2, -1.

-1.

-1.

y = sin x

(π/2, 1) (^) (5π/2, 1)

-0.

(0,0)^ (π, 0)

(2π, 0)

-1.

-1. (-π/2, -1) (^) (3π/2, -1)

If we continued the graph in both directions, we’d notice the following:

The domain is the set of all real numbers.

The range consists of all real numbers such that -1 ≤ sin x ≤ 1.

The functions f(x)The functions f(x) = sin x is an odd function since the graph is sin x is an odd function since the graph is

symmetric with respect to the origin. (f(-x) = -f(x) for every x in the

domain).

The period of the sine function is 2π.

The x-intercepts areThe x-intercepts are …., - 2 - 2ππ , --ππ , 0, 0 ππ , 2 2 ππ , 3 3 ππ , etc..etc

The y-intercepts is (0,0).

The maximum value is 1 and occurs at x = … -3π/2, π/2, 5π/2, etc..

The minimum value is -1 and occurs at x = … -π/2, 3π/2, 7π/2, etc..

Example 1 on p 176Example 1 on p. 176

How do you graph y= sin (x-π/4)?

Notice that this function is similar to y= sin x, with (x-π/4)

Instead of x. Therefore, this is just a horizontal shift to the RIGHT by π/

[show on TI-84]

y = cos x

  • π/3, 0.
    • π/6, 0.

0, 1. π/6, 0.

π/3, 0.500 5 π/3, 0.

11 π/6, 0.

2 π, 1. 13 π/6, 0.

0.500 7 π/3, 0.

-5π/6, -0.

-2π/3, -0.

  • π/2, 0.000 π/2, 0.

2 π/3, -0.

5 π/6, -0. π, -1.

7 π/6, -0.

4 π/3, -0.

3 π/2, 0.000 5 π/2, 0.

8 π/3, -0.

17 π/6, -0. -1.000^3 π,^ -1.

-0.

, ,

-1.

If we continued the graph in both directions, we’d notice the following: ThThe domain is the set of all real numbers. d i i h f ll l b The range consists of all real numbers such that -1 ≤ cos x ≤ 1. The functions f(x) = cos x is an even function since the graph is symmetric with respect to the y-axis. [f(-x) = f(x) for every x in the domain]. The period of the cosine function is 2π. The x-intercepts areThe x intercepts are …., - ππ/2/2, ππ/2 3/2, 3ππ/2 5/2, 5ππ/2 etc/2, etc… The y-intercepts is (0,1). The maximum value is 1 and occurs at x = … -2π, 0, 2π, etc.. The minimum value is -1 and occurs at x = … -π, π, 3π, etc..

y = Asin(ωx + h) + v or y = Acos(ωx + h) + v

|A| = amplitude = the biggest value of a periodically changing value. If A < 0 th h i fl t d th i

TRANSFORMATIONS

If A < 0, the graph is reflected on the x-axis. -|A| ≤ Asin x ≤ |A| and -|A| ≤ Acos x ≤ |A| [since sin x ≤ 1 and cos x ≤ 1] ω=Frequency = number of periods per time. The number of periods per second is measured in hertz (hz for short). We have a frequency of 1hz if a wave has exactly one period per second If we have 5 periods perif a wave has exactly one period per second. If we have 5 periods per second, we have a frequency of 5hz. Period = the length between two points which are surrounded by the same pattern. For sin and cos functions, period = 2π For tan and cot functions, period = π The new period of this function will either compress or stretch by a factor of 1/ ω. If the 0 < ω < 1, the new period is longer than the original function. If the ω > 1, the new period is shorter than the original function. For sin and cos: TT = period of new function = 2period of new function 2 ππ//ωω h = horizontal displacement. (+) If h > 0, graph is shifted to the LEFT. (-) If h < 0, graph is shifted to the RIGHT. v = vertical displacement (+) If v > 0, graph is shifted UP.

3

4

5

y = 5sin (.5x + pi/6)-1 y=sin x

( ) If v 0, graph is shifted UP. (-) If v < 0, graph is shifted DOWN.

3

0

1

2

π/ 6

π/ 3

π/ 2

2 π /

5 π /

3 π/ π (^2) 2 π 52 π / 3 π 72 π/ 4 π

y = − x

2 sin

Example 8 on p.184 π

From the formFrom the form yy = Asin(Asin(ωωx + h) + vx + h) + v

ω = -π/ (However we need ω > 0) Let’s remember that sin x is an odd function, where f(-x) = -f(x) So

y = − x x

2 sin

2 sin

Negative sign means sin function will be

The amplitude is |-2| = 2 so the largest value of y is 2. The period is T = 2π/ ω =

|A|=2 ω = π/

function will be reflected on the x-axis

h = 0, v = 0, so there is no horizontal or vertical displacement. Therefore, the period starts at 0 and ends at 4. sin x = 0 at x = π, so

0 ,when 2

sin ⎟= =

x x

⎝^2 ⎠

y = 2sin (-pi*x/2)

2

3

0

1

2

-4 -2 0 2 4 6 8 10 12

y = cos x y= sinx

1 000

-0.

π/2 π^3 π/2^2 π^5 π/2 3 π

-1.

-1.

Notice sin x and cos x are basically the same curves.Notice sin x and cos x are basically the same curves.

cos x is just sin x shifted to the left by π/2. Therefore,

cos x = sin (x + π/2).

Or alternatively, sin x is just cos x shifted to the right by π/2.

So

sin x = cos (xsin x = cos (x - ππ/2)/2).

Because of the similarity of cosine and sine curves,

these functions are often referred to as sinusoidal graphs.