The Unit Circle

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The Unit Circle
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The Unit Circle

• One of the most useful tools in trigonometry is
  the unit circle.
• It is a circle, with radius 1 unit, that is on
  the x-y coordinate plane.

sin
1
cos
The x-axis corresponds to the cosine function,
and the y-axis corresponds to the sine function.
The angles are measured from the positive x-axis
(standard position) counterclockwise. In order
to create the unit circle, we must use the
special right triangles below
45º
1
60º
1
30º
45º
30º -60º -90º
45º -45º -90º
The hypotenuse for each triangle is 1 unit.
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You first need to find the lengths of the other
sides of each right triangle...
45º
1
60º
1
30º
45º
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Now, use the corresponding triangle to find the
coordinates on the unit circle...
sin
(0, 1)
What are the coordinates of this point?
This cooresponds to (cos 30,sin 30)
(Use your 30-60-90 triangle)
(cos 30, sin 30)
cos
30º
(1, 0)
(1, 0)
(0, 1)
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Now, use the corresponding triangle to find the
coordinates on the unit circle...
sin
(0, 1)
What are the coordinates of this point?
(Use your 45-45-90 triangle)
(cos45, sin 45)
(cos 30, sin 30)
cos
45º
(1, 0)
(1, 0)
(0, 1)
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You can use your special right triangles to find
any of the points on the unit circle...
sin
(0, 1)
(cos45, sin 45)
(cos 30, sin 30)
cos
(1, 0)
(1, 0)
(Use your 30-60-90 triangle)
What are the coordinates of this point?
(0, 1)
(cos 270, sin 270)
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Use this same technique to complete the unit
circle on your own.
sin
(0, 1)
(cos45, sin 45)
(cos 30, sin 30)
cos
(1, 0)
(1, 0)
(cos 300, sin 300)
(0, 1)
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Unit Circle
(0, 1)
(-1, 0)
(1, 0)
0
(0, -1)
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Unit Circle
(0, 1)
30
30
(1, 0)
(-1, 0)
30
30
(0, -1)
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Unit Circle
(0, 1)
60
60
(1, 0)
(-1, 0)
60
60
(0, -1)
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Unit Circle
(0, 1)
45
45
(1, 0)
(-1, 0)
45
45
(0, -1)