Sine, Cosine, Tangent Rediscovered

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Sine, Cosine, Tangent Rediscovered

• Precalc 7.1

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Introduction

• Previously, we examined the graphs of sine and
  cosine and looked at ways of transforming them.
  Since the more we look, the better, lets review
  that now before exploring the mystic world of
  tangent.

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Sine revisited.
One period of sine

• y sin x
• Key features
• Period 2p
• Range -1,1
• Starts at (0, 0)

2p
p/2
3p/2
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Cosine revisited.
One period of cosine

• y cos x
• Key features
• Period 2p
• Range -1,1

p/2
3p/2
-p/2
Same as the sine graph except shifted left p/2!
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Tangent time!

• y tan x
• Key features
• Range all reals
• Period p
• Funky fun!

p/2
-p/2
-3p/2
3p/2
Asymptotes x p/2 pk
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This nThat

• Sine and cosine have lots of pretty points that
  can be graphed. Tangentwell, not so much. With
  tangent, youre mostly making a sketch. So the
  key is to get the asymptotes right along with the
  center point.

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Even vs Odd

• Remember
• Even symmetric with respect to y-axis
• Odd symmetric with respect to origin (180
  rotation)
• So which trig functions are even and which are
  odd?
• Sine and tangent are odd. Easy to prove f(-x)
  -f(x) for any value of x!
• Cosine is even. Also easy f(-x) f(x) for any
  value of x.

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Transformation Central!

• These graphs like to move, so lets boogie.
• Consider something fun like
• y 3 sin (x p/2) - 2. (Its fun dagnabit!)
• Dude, its just a transformation of the original
  sine graph.
• The 3 triples the range to -3, 3.
• The -2 shifts the graph down to so the new
  range is -5, 1.
• The p/2 shifts the graph LEFT p/2. (Think of
  the graph as going from -p/2 to 3p/2 instead of
  from 0 to 2p! Thinking in terms of one period
  makes stuff easier.)

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Did ya get it?
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Strategery.

• BEFORE putting pencil to paper, figure out
• 1) How has the range changed?
• 2) How has the period changed? (Weve havent
  talked much about this yet.)
• 3) How is the graph shifted?
• THEN graph one period of the function using your
  new information.
• THEN graph the rest of the pattern.

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Go ballistic.

• Graph WITHOUT a calculator
• y -2 cos (x - p) 1
• The 2 effects the range -2, 2
• The 1 shifts range up one -1, 3
• The -p shifts us RIGHT p.
• The - out in front flips the graph upside down.

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Lets take a look!

• Heres one period from p to 3p.
• Heres the whole thing.