Download Understanding Sine and Cosine: Period, Amplitude, and Transformations and more Study Guides, Projects, Research Trigonometry in PDF only on Docsity!
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
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.
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.
