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Trigonometric Functions
Sine and Cosine Functions
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f(x) = sin x and f(x) = cos x
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f(x) = sin x and f(x) = cos x
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f(x) = sin x & two important ideas
Period
Am
plitudePeriod
Period means how many degrees in one cycle.
Amplitude means the distance from the centre to the maximum or minimum, OR (max + min) ÷ 2
Am
plitude
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f(x) = sin x
Period
Period = 360º
Amplitude = 1
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Now we will investigate
f(x) = A sin Bx + C
How do A, B and C affect the shape of the graph?
Note: It is exactly the same for sine and cosine, so we will stick just to sine for the start.
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f(x) = sin x & f(x) = sin x + 3
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f(x) = sin x + 3 & f(x) = sin x – 2
So C moves the curve up and down
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f(x) = sin x & f(x) = sin 2x
Period = 180º
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f(x) = sin x & f(x) = sin 3x
So B changes the period;
the period of the function is (360º ÷ B)
Period
= 120º
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f(x) = sin x & f(x) = 2 sin x
Amplitude = 2
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f(x) = sin x & f(x) = 4 sin x
Amplitude = 4
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f(x) = sin x & f(x) = -1 sin x
Amplitude = 1
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f(x) = sin x & f(x) = -3 sin x
Amplitude = 3
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f(x) = sin x & f(x) = A sin x
The A gives the amplitude of the function.
A negative value means the graph goes down – up, not up – down.
A = 4
A = -3
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f(x) = A sin Bx + C
A = amplitude
Note: It is exactly the same for sine and cosine.
The difference is the where it crosses the y-axis.
B = 360º ÷ period
C = vertical shift
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What is the equation of this function?
Amplitude = 2
Period = 120º
Vertical shift = -1
f(x) = 2 sin 3x – 1
so, A = 2
so, B = 3
so, C = -1
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What is the equation of this function?
Amplitude = 4,
going down-up
Period = 720º
Vertical shift = 1
f(x) = -4 sin ½x + 1
so, A = -4
so, B = 0.5
so, C = 1
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What is the equation of this function?
Amplitude = 2.5
Period = 240º
Vertical shift = 2
f(x) = 2.5 sin 1.5x + 2
so, A = 2.5
so, B = 1.5
so, C = 2