The Unit Circle

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The Unit Circle
Radius 1
Radius 1
Radius 1
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A Reference Triangle
The angle relative to the x-axis is shown as
ALPHA (a).
The acute angle formed by the radius and the
x-axis in that quadrant is THETA (?)
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Now imagine it dynamically moving around the
circle to any point that you want.
Imagine a point ANYWHERE on the unit circle
Here are several examples of reference triangles
in each quadrant.
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The Whole VISUAL Analogy
SKY
DISCLAIMER No real animals were killed or harmed
in the production of this film. However, a few
cats were possibly roughed up, as their unwanted
assistance became annoying.
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OCEAN
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Meet the Four Families on the Block
Meet the Quadrantal Family the weird ones - on
the Axes
Look for patterns listen to the tips from your
teacher

• Cut the circle into four pieces each equaling
  90 degrees
• No actual Reference Triangles are formed
• or one can visualize the collapsed triangles
• The only family with 5 members
• 0 and 2p 0 and 360 overlap as the start and end
  points
• The coordinates of 0 and 1 make the sines and
  cosines easy

Lets take a look
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The Quadrantal Family
(0, 1)
Around the circle in DEGREES
Around the circle in RADIANS
Around the circle in COORDINATES
(1, 0)
(-1, 0)
(0, -1)
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Now Lets Meet the Square Family
Look for patterns listen to the tips from your
teacher

• Cut the circle into EIGHT pieces each equaling
  45 degrees
• Each Reference Triangle is a Right Isosceles
  Triangle
• Use only the odd multiples of
• The numerator values count 1-3-5-7
• All coordinates are
• Connect the points and you get a perfect square

Lets take a look
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Around the circle in DEGREES
Around the circle in RADIANS
Around the circle in COORDINATES
How far L/R is the bird? this is the cosine of
the angle.
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How high in the air is the bird? this is the
sine of the angle.
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Now Lets Meet the Wide Rectangle Family
Look for patterns listen to the tips from your
teacher

• Cut the circle into TWELVE pieces each equaling
  30 degrees
• Each Reference Triangle is a laying down
  30-60-90 Triangle
• The numerator values count 1-5-7-11
• All x coordinates are , and all y
  coordinates are
• Connect the points and you get a 1 by root 3
  rectangle

Lets take a look
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Around the circle in RADIANS
Around the circle in DEGREES
How deep in the water is the fish? this is the
sine of the angle.
How far L/R is the fish? this is the cosine of
the angle.
Around the circle in COORDINATES
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Now Lets Meet the Tall Rectangle Family
Look for patterns listen to the tips from your
teacher

• Cut the circle into SIX pieces each equaling 60
  degrees
• Each Reference Triangle is an upright 30-60-90
  Triangle
• The numerator values count 1-2-4-5
• All x coordinates are , and all y
  coordinates are
• Connect the points and you get a root 3 by 1
  rectangle

Lets take a look
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Around the circle in RADIANS
Around the circle in DEGREES
How deep in the water is the fish? this is the
sine of the angle.
Around the circle in COORDINATES
How far L/R is the fish? this is the cosine of
the angle.
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