The Unit Circle

1
The Unit Circle

• Part II
• (With Trig!!)
• MSpencer

2
Multiples of 90,
180, ?
360, 2?
0, 0
3
The Quadrants (with Angles)
180, ?
360, 2?
0, 0
4
The Unit Circle
Remember it is called a unit circle because the
radius is one unit. So lets add in ordered pairs
to the unit circle.
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Multiples of 90,
(0, 1)
r 1
180, ?
0, 0
(1, 0)
(?1, 0)
r 1
r 1
r 1
(0, ?1)
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45,
7
45,
Lets review this triangle from geometry.
Opposite the congruent, 45 angles are congruent
sides.
These sides are the legs of the right triangle.
So the triangle is an isosceles right triangle.
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45,
Lets call the two congruent legs s.
9
45,
Lastly, now remember that the hypotenuse is the
radius of the unit circle, which means it must
equal one. Solve for s.
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45,
1
The distance across the bottom side of the
triangle represents the x-coordinate while the
right, vertical side represent y.
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Signs and Quadrants
The signs of each ordered pair follow the signs
of x and y for each quadrant.
Q I (, )
Q II (?, )
180, ?
0, 0
Q III (?, ?)
Q IV (, ?)
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Multiples of 45,
13
60,
14
60,
Lets review this triangle from geometry. Call
the the smallest side opposite 30 s.
2s
The hypotenuse is twice the smallest side, or 2s.
s
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60,
The hypotenuse is the radius of the unit circle,
which means it must equal one. Solve for s.
2s 1
s
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60,
Notice that since the triangle is taller than it
is wide, that the y-coordinate is larger than the
x-coordinate.
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Multiples of 60,
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30,
Notice this is the same special right triangle as
for 60 except the x and y coordinates are
switched.
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Multiples of 30,
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Ordered Pairs and Trig
From geometry, recall SOHCAHTOA, which defines
sine, cosine, and tangent. sine (Sin)
cosine(Cos) tangent (Tan)
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Ordered Pairs and Trig
Cos 30
cos 30
Notice that the cosine of the angle is simply the
x-coordinate!
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Ordered Pairs and Trig
Sin 30
sin 30
Notice that the sine of the angle is simply the
y-coordinate!
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Ordered Pairs Cosine Sine
(cos ?, sin ?)
(x, y)
?
And this is true for ANY angle, often called
?. cos ? x sin ? y
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Signs for Cosine and Sine
The signs of cosine and sine follow the
signs of x and y in each quadrant.
Q I (, )
Q II (?, )
180, ?
0, 0
Q III (?, ?)
Q IV (, ?)
So in QII, for instance, cosine is negative while
sine is positive.
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The Whole Unit Circle Together (Grouped)
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The Whole Unit Circle Together (In Ascending
Order)