2 Unit Bridging Course – Day 10
Circular Functions II – The general sine function
Clinton Boys
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The general sine function
We’re now going to consider variations on the sine function.
This is similar to the fact that y = x is a linear function (perhaps
the prototypical linear function), but that
ax + by + c = 0
represents the most general form of a linear function.
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The general sine function
We’re now going to consider variations on the sine function.
This is similar to the fact that y = x is a linear function (perhaps
the prototypical linear function), but that
ax + by + c = 0
represents the most general form of a linear function.
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Amplitude
First let’s think about what happens when we multiply the sine
function by some number A.
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Amplitude
First let’s think about what happens when we multiply the sine
function by some number A.
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Amplitude
First let’s think about what happens when we multiply the sine
function by some number A.
y = sin x
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Amplitude
First let’s think about what happens when we multiply the sine
function by some number A.
y = 2 sin x
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Amplitude
First let’s think about what happens when we multiply the sine
function by some number A.
y = 3 sin x
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Amplitude
First let’s think about what happens when we multiply the sine
function by some number A.
y = 4 sin x
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Amplitude
First let’s think about what happens when we multiply the sine
function by some number A.
y = 4 sin x
y = 3 sin x
y = 2 sin x
y = sin x
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Amplitude
Ordinarily, the graph of y = sin x is always between y = 1 and
y = −1.
When we multiply by A, the graph of y = A sin x is always
between y = A and y = −A.
A is called the amplitude of the function y = A sin x.
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Amplitude
Ordinarily, the graph of y = sin x is always between y = 1 and
y = −1.
When we multiply by A, the graph of y = A sin x is always
between y = A and y = −A.
A is called the amplitude of the function y = A sin x.
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Amplitude
Ordinarily, the graph of y = sin x is always between y = 1 and
y = −1.
When we multiply by A, the graph of y = A sin x is always
between y = A and y = −A.
A is called the amplitude of the function y = A sin x.
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Period
Now let’s think about what happens when we change x to ωx
for some number ω.
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Period
Now let’s think about what happens when we change x to ωx
for some number ω.
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Period
Now let’s think about what happens when we change x to ωx
for some number ω.
y = sin x
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Period
Now let’s think about what happens when we change x to ωx
for some number ω.
y = sin(2x)
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Period
Now let’s think about what happens when we change x to ωx
for some number ω.
y = sin(x/2)
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Period
Now let’s think about what happens when we change x to ωx
for some number ω.
y = sin(3x)
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Period
Now let’s think about what happens when we change x to ωx
for some number ω.
y = sin x
y = sin(2x)
y = sin(x/2)
y = sin(3x)
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Period
If we change y = sin x to y = sin(ωx), the effect is to stretch or
compress the graph horizontally.
If ω > 1, then the graph becomes compressed – we need to
now fit in the full sine curve ω times where we previously fitted it
in once.
If ω < 1 then the graph becomes stretched – we now have 1/ω
times as much space to fit in a full sine curve.
The number
2π
is called the period of the function y = sin(ωx).
ω
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Period
If we change y = sin x to y = sin(ωx), the effect is to stretch or
compress the graph horizontally.
If ω > 1, then the graph becomes compressed – we need to
now fit in the full sine curve ω times where we previously fitted it
in once.
If ω < 1 then the graph becomes stretched – we now have 1/ω
times as much space to fit in a full sine curve.
The number
2π
is called the period of the function y = sin(ωx).
ω
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Period
If we change y = sin x to y = sin(ωx), the effect is to stretch or
compress the graph horizontally.
If ω > 1, then the graph becomes compressed – we need to
now fit in the full sine curve ω times where we previously fitted it
in once.
If ω < 1 then the graph becomes stretched – we now have 1/ω
times as much space to fit in a full sine curve.
The number
2π
is called the period of the function y = sin(ωx).
ω
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Sketching sine curves
If we are given a general sine curve y = A sin(ωx), we can
sketch it without using calculus.
2π
,
ω
this is enough information to sketch the curve, since we know
that the basic shape of the curve is the same as for y = sin x.
Indeed, if we know that the amplitude is A, and the period is
We just need to change the height of the curve according to its
amplitude A, and stretch or compress the curve horizontally
according to its period 2π/ω.
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Sketching sine curves
If we are given a general sine curve y = A sin(ωx), we can
sketch it without using calculus.
2π
,
ω
this is enough information to sketch the curve, since we know
that the basic shape of the curve is the same as for y = sin x.
Indeed, if we know that the amplitude is A, and the period is
We just need to change the height of the curve according to its
amplitude A, and stretch or compress the curve horizontally
according to its period 2π/ω.
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Sketching sine curves
If we are given a general sine curve y = A sin(ωx), we can
sketch it without using calculus.
2π
,
ω
this is enough information to sketch the curve, since we know
that the basic shape of the curve is the same as for y = sin x.
Indeed, if we know that the amplitude is A, and the period is
We just need to change the height of the curve according to its
amplitude A, and stretch or compress the curve horizontally
according to its period 2π/ω.
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Example
Example
Sketch the graph of the function y = 2 sin(3x).
The amplitude is 2.
The period is
2π
(i.e. the graph repeats itself every 2π/3).
3
2
2π/3
2
−2π
3
π
3
−π
3
2π
3
−2
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Example
Example
Sketch the graph of the function y = 2 sin(3x).
The amplitude is 2.
The period is
2π
(i.e. the graph repeats itself every 2π/3).
3
2
2π/3
2
−2π
3
π
3
−π
3
2π
3
−2
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Example
Example
Sketch the graph of the function y = 2 sin(3x).
The amplitude is 2.
The period is
2π
(i.e. the graph repeats itself every 2π/3).
3
2
2π/3
2
−2π
3
π
3
−π
3
2π
3
−2
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Example
Example
Sketch the graph of the function y = 2 sin(3x).
The amplitude is 2.
The period is
2π
(i.e. the graph repeats itself every 2π/3).
3
2
2π/3
2
−2π
3
π
3
−π
3
2π
3
−2
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Practice questions
Practice questions
For each of the following sine curves, write down the amplitude,
the period, and sketch the curve.
(i) y = 3 sin(x/2)
(ii) y = 2 sin(x/3)
(ii) y = (1/2) sin(2x)
(iv) y = 4 sin(πx).
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Practice questions
Answers
(i) amplitude 3, period 4π
(ii) amplitude 2, period 6π
(iii) amplitude 21 , period π
(iv) amplitude 4, period
2π
π
= 2.
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