Definition 3: Trigonometric Functions: The Unit Circle 3'4

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Definition 3 Trigonometric FunctionsThe Unit
Circle3.4

• JMerrill, 2009
• Contributions from DDillon

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Recall Definitions of Trig Functions

• Definition 1 involved the ratios of 2 sides of a
  triangle (SOH CAH TOA)
• Definition 2 dealt with ratios using x- and
  y-coordinates and the distance from the origin to
  a point (using xs, ys, and rs)

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The Unit Circle
This circle has radius of 1. It is centered at
the origin. Endpoints are labeled as
(0,1)
(1, 0)
(-1, 0)
This is the standard that we use. All our
function values are based on this standard.
(0, -1)
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Definition 3 The Unit Circle

• Let (x, y) be any point on the unit circle. If ?
  is the central angle that has the same measure as
  the arc length from the point (1,0) along the
  circumference to the point (x, y), then
• The coordinates of the points along the unit
  circle can be written (cos?, sin?).

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Trig Function Values of Quadrantal Angles
xs are cosines ys are sines
(0,1)
0

• sin 180º _____
• 2. cos 90º _____
• 3. cot 270º _____
• 4. tan 90º _____
• 5. csc _____
• 6. Sec _____

(1, 0)
(-1, 0)
0
0
undef
1
1
(0, -1)
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Recall 45-45-90 Triangles
In any 45-45-90 triangle, the sides are in the
ratio 1 1 v2.
sin 45 v2/2 cos 45 v2/2 tan 45 1
45
1
v2/2
v2/2
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The Unit Circle
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Recall 30-60-90 Triangles
In any 30-60-90 triangle, the sides are in the
ratio 1 2 v3
sin 60 v3/2 cos 60 1/2 tan 60 v3 sin
30 1/2 cos 30 v3/2 tan 30 v3/3
30
1
v3/2
1/2
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The Unit Circle
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Trig Function Values
v2/2
1/2

• sin 30º _______ 4. cos p/4 _______
• tan p/3 _______ 5. sec p/6 _______
• 3. sin p _______ 6. cot p/2 _______

v3
2v3/3
0
0