Mat 161 PreCalculus

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Mat 161 - PreCalculus

• Sine and Cosine Functions
• Sections 5.5

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Sine The Function
Let f(x) sin (x). Then we know that The domain
is all real numbers and the range is -1, 1.
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Sine The Function
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Properties of f(x) sin(x)

• Domain (-8, 8)
• Range -1, 1
• Period 2p
• y-intercept 0
• x-intercepts 0, p, 2p (all integer multiples
  of p)
• Odd, symmetric about the origin

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Cosine The Function
Let f(x) cos (x). Then we know that The domain
is all real numbers and the range is -1, 1.
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Cosine The Function
1
4p
2p
3p
p
-1
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Properties of f(x) cos(x)

• Domain (-8, 8)
• Range -1, 1
• Period 2p
• y-intercept 1
• x-intercepts p/2, 3p/2 (all odd multiples of
  p/2)
• Even, symmetric about the y-axis

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General Sine and Cosine Curves

• The General Form of any Sine and Cosine functions
  is
• Y Asin(Bx C) D
• and
• y Acos(Bx C) D
• To understand their graphs we need to remember
  some transformation laws.

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Recall

• Given the graph of y f(x) and real number
  c, we know that
• y f(xc) is a shift horizontally
• y f(x) c is a shift vertically
• Y -f(x) is a reflection about the x-axis
• Y f(-x) is a reflection about the y-axis
• Y af(x) is a stretch/shrinking vertically
• Y f(ax) is a stretch/shrinking horizontally

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Properties of general Sine and Cosine

• Based on this, we have that for
• y Asin(Bx C) D or y Acos(Bx C) D
• Amplitude A
• Period 2p/B
• Phase shift C/B
• Vertical shift D

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TheTrick ?!

• Know the basic 5 points for each function
• Find the amplitude
• Find the period
• Find where the graph begins and ends
• Plot the start and the finish and fill in the
  rest ?!

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Graph f(x) 2sin(x)

• Amplitude
• Period
• Start
• Finish

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Graph f(x) -?cos(x)

• Amplitude
• Period
• Start
• Finish

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Graph f(x) cos(-3x)

• Amplitude
• Period
• Start
• Finish

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Graph f(x) sin(xp/3)

• Amplitude
• Period
• Start
• Finish

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Graph f(x) -2sin(x-p/2)

• Amplitude
• Period
• Start
• Finish

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Graph f(x) -2cos(px-p)

• Amplitude
• Period
• Start
• Finish

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Unit Circle Trigonometry

• Reference
• Algebra and Functions
• Blitzer
• 3rd Edition