Download Trigonometry Notes on the Sine and Cosine Functions and Their

Trigonometry
Notes on the Sine and Cosine Functions and Their Graphs.
The Sine Function: First remember that the principle definition of the sine function is that
sin ( θ ) = the y-coordinate on the unit circle at an angle in standard position, θ .
Also, remember that any number can be an angle since we can allow for multiple rotations and direction
of rotation. Thus, as we keep increasing θ, we will just keep rotating around the unit circle, and the
y-coordinate , sin(θ), will just keep changing between 1 and -1 and back again. Also, because an angle in
radians is really just an unitless number, we actually evaluate sine at a number that is not necessarily an
angle as in sin(x) instead of sin(θ). See figures to the right and below. Be careful not to confuse the x on
the unit circle and the x in the graph of the sine function.
y =1
y=0
y=0
y = -1
θ
Next let’s be more precise. Using angles from -2π to 2π that have basic reference angles, we get the following.
x
sin(x)
x
sin(x)
x
sin(x)
x
sin(x)
#
-2π
11π
6
#
0
1
2
-π
5π
6
0
1
2
0
π
6
0
1
2
π
7π
6
0
1
2
7π
4
5π
3
3π
2
4π
3
2
2
2
2
3
2
π
4
π
3
π
2
2π
3
5π
4
4π
3
3π
2
5π
3
2
2
1
2
3π
4
5π
6
-
5π
4
7π
6
-
3
2
1
3
2
2
2
1
2
3π
4
2π
3
π
2
π
3
-
π
4
π
6
-
-
2
2
-
3
2
-1
-
3
2
1
3
2
2
2
1
2
7π
4
11π
6
2π
#
-
2
2
-
3
2
Graph of y = sin(x) on -2π < x < 2π
-1
-
3
2
2
2
1
2
0
#
The shape of this graph is called a sinusoidal wave, or just sine wave.
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #1 of 10
Because we can keep going around the unit circle, the value of sin ( x ) will oscillate between -1 and 1 at regular intervals, every
rotation. Functions like this are said to be periodic.
Periodic Functions
The function, f(x), is periodic, if and only
if, there exists a real number, p, such that
f(x) = f(x + p) for all x in domain.
p is called the period on the function.
The period can also refer to the section of
the graph that is repeated.
Hence, since sin(x) will repeat every full rotation, its period is 2π. Also, the graph is just repeating what is in the interval [ 0,2π] , the
interval [ 0,2π] is called the basic period y = sin(x).
Another characteristic of the graph of y = sin(x) is its amplitude.
Amplitude
The amplitude of a sinusoidal wave is the
maximum displacement from the vertical
center of the graph.
Thus, y = sin(x) has an amplitude of 1.
Summarizing what’s above.
The Basic Sine Function
y = sin(x)
Domain = ( -∞,∞ )
Range = [ -1,1]
Amplitude = 1
Period = 2π
Basic Period = [ 0,2π]
Odd function: Symmetric to origin
sin(-x) = -sin(x)
Graphing Criterion:
At least 1 period,
5 plotted points including:
maximum point,
minimum point,
the 3 x-intercepts.
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #2 of 10
Changing the Amplitude and Vertical Reflections:
If we want to change the amplitude of a sine wave, we need to stretch or shrink the graph vertically. Remember from College
Algebra that y = a ⋅ f(x) where a > 0 performs a vertical stretch or shrink of the graph.
Example #1: Graph y = 5sin (x) .
►
Since the graph is vertically stretched by a factor of 5, the amplitude = 5. Nothing else changes.
Amplitude = 5
Period = 2π
□
Example #2: Graph y = -3sin (x) .
►
This one has both a vertical stretch by a factor of 3 and a vertical reflection caused by the negative. Thus, the amplitude is 3 and the
graph will be in a way upside-down.
Amplitude = 3
Vertical Reflection: Yes
Period = 2π
□
Example #3: Graph y =
sin (x)
.
3
►
1
3
Vertical Reflection: No
Period = 2π
Amplitude =
□
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #3 of 10
Changing the Period:
Example #4: Graph y = sin (2x) .
►
For 2x to go though the basic period, [ 0,2π] ,
0 ≤ 2x ≤ 2π
0 2x 2π
≤
≤
2 2
2
0≤x≤π
Thus, when x goes from 0 to π, 2x will go from 0 to 2π, and the graph will go through an entire period. Therefore, the period of
y = sin (2x) is π.
Since the basic period of sine starts and ends at x-intercepts and has an x-intercept halfway between them, y = sin (2x) will have
π
x-intercepts at 0, , and π .
2
On the basic period of sine has a maximum halfway between the 1st and 2nd x-intercepts, and a minimum halfway between the 2nd
π
3π
and 3rd x-intercepts, the maximum will occur when x = , and the minimum will be when x =
.
4
4
Amplitude = 1
Vertical Reflection: No
Period = π
Basic Period moved to [ 0, π]
□
Example #5: Graph y = sin (x/4) .
►
x
≤ 2π
4
x
4 ⋅ 0 ≤ 4 ⋅ ≤ 4 ⋅ 2π
4
0 ≤ x ≤ 8π
0≤
Amplitude = 1
Vertical Reflection: No
Period = 8π
Basic Period moved to [ 0,8π]
□
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #4 of 10
Phase Shifting the Graph:
Remember that y = f (x − h) is the graph of y = f (x) shifted horizontally. With graphs of trigonometric functions, a horizontal shift
is called a phase shift.
Example #6: Graph y = sin (x − π) .
►
The graph will be the basic graph of y = sin (x) shifted to the right π units.
You can also look at like we did with the change in period.
0 ≤ x − π ≤ 2π
0 + π ≤ x − π + π ≤ 2π + π
π ≤ x ≤ 3π
Thus the basic period has moved to [ π,3π]
Amplitude = 1
Vertical Reflection: No
Period = 2π
Phase Shift = π
Basic Period moved to [ π,3π]
□
⎛ π⎞
Example #7: Graph y = sin ⎜ x+ ⎟ .
⎝ 2⎠
►
π
≤ 2π
2
π
π π 4π π
−
0− ≤ x+ − ≤
2
2 2 2 2
π
3π
- ≤x≤
2
2
0≤x+
Amplitude = 1
Vertical Reflection: No
Period = 2π
π
Phase Shift = 2
⎡ π 3π ⎤
Basic Period moved to ⎢ - , ⎥
⎣ 2 2⎦
□
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #5 of 10
Shifting the Graph Vertically:
Remember that y = f (x) + k is the graph of y = f (x) shifted vertically. This will move the vertical center of the graph, sometimes
called the baseline. For the basic graph of y = sin ( x ) , the baseline is the x-axis.
Example #8: Graph y = sin ( x ) + 7 .
►
The graph will be the graph of y = sin ( x ) moved up 7 units.
Amplitude = 1
Vertical Reflection: No
Period = 2π
Phase Shift = 0
Basic Period moved to [ 0, 2π]
Vertical Shift = 7
□
Example #9: Graph y = sin ( x ) − 5 .
►
The graph will be the graph of y = sin ( x ) moved down 5 units.
Amplitude = 1
Vertical Reflection: No
Period = 2π
Phase Shift = 0
Basic Period moved to [ 0, 2π]
Vertical Shift = -5
□
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #6 of 10
Summary of the Graph of the General Sine Function:
For the general y = a ⋅ sin(bx + c) + d where b>0, we get
0 ≤ bx + c ≤ 2π
-c ≤ bx ≤ 2π − c
c
2π − c
≤x≤
b
b
c
⎛ 2 π − c ⎞ ⎛ c ⎞ 2π − c c 2π
Thus, the phase shift = - , and the period = ⎜
+ =
⎟ −⎜- ⎟ =
b
b
b b
⎝ b ⎠ ⎝ b⎠
-
The General Sine Function
y = a sin(bx+c)+d where b > 0
Domain = ( -∞,∞ )
Range = ⎡⎣d − a ,d+ a ⎤⎦
Amplitude = a
Period =
2π
b
Phase Shift = -
c
b
⎡ c 2π − c ⎤
Basic Period moved to = ⎢ - ,
b ⎥⎦
⎣ b
Vertical Shift = d
Graphing Criterion:
At least 1 period,
5 plotted points including:
maximum point,
minimum point,
the 3 baseline-intercepts.
Example #10: Graph y = -2sin ( 3x − π ) + 1 . Identify amplitude, period, phase shift and vertical shift.
►
Using the formulas above.
Amplitude = -2 = 2
Vertical Reflection: Yes
2π
Period =
3
PhaseShift = -
-π π
=
3 3
⎡ -π 2π − ( -π ) ⎤ ⎡ π 3π ⎤ ⎡ π ⎤
Basic Period moved to ⎢ - ,
⎥ = ⎢ , ⎥ = ⎢ , π⎥
3
⎣ 3
⎦ ⎣3 3 ⎦ ⎣3 ⎦
VerticalShift = 1
□
Many of you might find the formulas too much to remember. In that case, you can still use the method of solving the 3 part
inequality to get where the basic period has moved to, and then divide that interval up into 4 parts to get the max, min, and 3rd baseline
intercept.
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #7 of 10
Example #11: Graph y = -3sin ( -πx − 8π ) − 4 . Identify amplitude, period, phase shift and vertical shift.
►
Since this one has a negative coefficient on x, 1st factor out the negative and use the negative number identify for sine to either drop
the negative or bring it out front. In sine’s case, sin(-x) = -sin(x) . Thus, we move the negative out front.
y = -3sin ( -πx − 8π ) − 4
y = -3sin ⎡⎣ - ( πx+8π ) ⎤⎦ − 4
y = - ( -3sin [ πx+8π]) − 4
y = 3sin ( πx+8π ) − 4
We now focus on this last form to get the graph.
Now using the inequality:
0 ≤ πx+8π ≤ 2π
-8π ≤ πx ≤ -6π
-8 ≤ x ≤ -6
Thus, the basic period has moved to [-8,-6]. The 3 baseline-intercept will be when x = -7, the center of the interval. Also, since there
15
13
is no vertical reflection, the max will be when x = - , halfway between -8 and -7, and the min will be when x = - , halfway
2
2
between -7 and -6.
rd
Amplitude = 3 = 3
Vertical Reflection: No
Period = ( -6 ) − ( -8 ) = 2
PhaseShift = -8
Basic Period moved to [ -8, -6]
VerticalShift = - 4
□
It doesn’t matter which way you get the information, but you do have to know some way to find it and the 5 points to plot.
The Cosine Function: ⎡ ⎛
π ⎞⎤
π⎞
⎛π
⎞
⎛
For the graph of y = cos ( x ) , first note that y = cos ( x ) = sin ⎜ − x ⎟ = sin ⎢- ⎜ x − ⎟ ⎥ = -sin ⎜ x − ⎟ . Thus, the graph of
2
2
2⎠
⎝
⎠
⎠⎦
⎝
⎣ ⎝
y = cos ( x ) is also a sinusoidal wave and is just transformations of the basic sine function. Specifically, it has a vertical reflection and
a phase shift of
π
.
2
Usually we want to keep the basic period of a function as close to 0 as possible. Thus, we will use the same basic period of [ 0, 2π]
even for cosine. Hence, the basic period looks different than sine since we are using a different part of the wave.
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #8 of 10
The Basic Cosine Function
y = cos(x)
Domain = ( -∞,∞ )
Range = [ -1,1]
Amplitude = 1
Period = 2π
Basic Period = [ 0,2π]
Odd function: Symmetric to y-axis
sin(-x) = sin(x)
Graphing Criterion:
At least 1 period,
5 plotted points including:
the 2 maximum point,
minimum point,
the 2 x-intercepts.
Since we set the basic period back to [ 0,2π ] , and the graph of cosine is also a sinusoidal wave, the formulas and methods for
graphing a cosine function are the same as for graphing a sine function.
The General Cosine Function
y = a cos(bx+c)+d where b > 0
Domain = ( -∞,∞ )
Range = ⎡⎣ d − a ,d+ a ⎤⎦
Amplitude = a
Period =
2π
b
Phase Shift = -
c
b
⎡ c 2π − c ⎤
Basic Period moved to = ⎢ - ,
b ⎥⎦
⎣ b
Vertical Shift = d
Graphing Criterion:
At least 1 period,
5 plotted points including:
the 2 maximum point,
minimum point,
the 2 baseline-intercepts.
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #9 of 10
Example #12: Graph y = 5cos ( -2x+π ) − 3 . Identify amplitude, period, phase shift and vertical shift.
►
1st deal with the negative on x. Since cos(-x) = cos(x) , we’ll drop the negative.
y = 5cos ⎡⎣ - ( 2x − π ) ⎤⎦ − 3
y = 5cos ( 2x − π ) − 3
Using the formulas above,
Amplitude = 5 = 5
Vertical Reflection: No
2π
Period =
=π
2
-π π
PhaseShift = - =
2 2
⎡ -π 2π − ( -π ) ⎤ ⎡ π 3π ⎤
Basic Period moved to ⎢ - ,
⎥=⎢ , ⎥
2
⎣ 2
⎦ ⎣2 2 ⎦
VerticalShift = -3
□
⎛ x+π ⎞
Example #13: Graph y = - cos ⎜
⎟ + 4 . Identify amplitude, period, phase shift and vertical shift.
⎝ 2 ⎠
►
Using the inequality this time,
x+π
≤ 2π
0≤
2
0 ≤ x + π ≤ 4π
-π ≤ x ≤ 3π
Amplitude = -1 = 1
Vertical Reflection: Yes
Period = ( 3π ) − ( -π ) = 4π
PhaseShift = -π
Basic Period moved to [ -π,3π]
VerticalShift = 4
□
Applications: The Applications are done in a different set of notes.
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Notes on the Sine and Cosine Functions and Their Graphs.
Page #10 of 10