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2.6 Graphs of the Sine and Cosine Functions
2 π =B13+PI()/6 =SIN(B14)
11 π/6 =B12+PI()/6 =SIN(B13)
5 π/3 =B11+PI()/6 =SIN(B12)
3 π/2 =B10+PI()/6 =SIN(B11)
4 π/3 =B9+PI()/6 =SIN(B10)
7 π/6 =B8+PI()/6 =SIN(B9)
π =B7+PI()/6 =SIN(B8)
5 π/6 =B6+PI()/6 =SIN(B7)
2 π/3 =B5+PI()/6 =SIN(B6)
π/2 =PI()/2 =SIN(B5)
π/3 =PI()/3 =SIN(B4)
π/6 =PI()/6 =SIN(B3)
0 0 =SIN(B2)
x x y = sin x
If we graph y = sin x by
plotting points, we see
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
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
graph a more complete
graph of y= sin x, we
repeat this period in each
direction.
y = sin x
2 π 6.283 0.
11 π/6 5.76 -0.
5 π/3 5.236 -0.
3 π/2 4.712 -1.
4 π/3 4.189 -0.
7 π/6 3.665 -0.
π 3.142 0.
5 π/6 2.618 0.
2 π/3 2.094 0.
π/2 1.571 1.
π/3 1.047 0.
π/6 0.524 0.
x x y = sin x
y = sin x
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) = 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 are …., - 2π, -π, 0, π, 2π, 3π, 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..
(-π/2, -1)
(π/2, 1)
(π, 0)
(3π/2, -1)
(0,0)
(2π, 0)
(5π/2, 1)
Example 1 on p. 155
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-83]
y = cos x
-5π/6, -0.
-2π/3, -0.
0, 1. π/6, 0.
π/3, 0.
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.
5 π/3, 0.
11 π/6, 0.
2 π, 1. 13 π/6, 0.
7 π/3, 0.
5 π/2, 0.
8 π/3, -0.
17 π/6, -0. 3 π, -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 ≤ 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 are …., - π/2, π/2, 3π/2, 5π/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..
0
1
2
3
4
5
y = 5sin (.5x + pi/6)-1 y=sin x
y = Asin(ωx + h) + v or y = Acos(ωx + h) + v
|A| = amplitude = the biggest value of a periodically changing value.
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 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:
T = period of new function = 2π/ ω
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.
(-) If v < 0, graph is shifted DOWN.
TRANSFORMATIONS
π/ 6
π/ 3
π/ 2
2 π /
5 π /
3 π/ π (^2) 2 π 5 π/ 2 3 π^4 π
7 π/ 2
Now you try
y = cos x y= sinx
π/2 π^3 π/2^2 π^5 π/2 3 π
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 (x - π/2).
Because of the similarity of cosine and sine curves,
these functions are often referred to as sinusoidal graphs.
Homework
p. 166-
#11* , 23,25,27, 33,35, 39*,43, 53,
63, 67, 69, 81
