Introduction to the Unit Circle and Right Triangle Trigonometry

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Introduction to the Unit Circle andRight
Triangle Trigonometry

• Presented by,
• Ginny Hayes
• Space Coast Jr/Sr High

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Draw the circle

• Label the x- and y-intercepts. Your circle
  should look like this

(0,1)
(1,0)
(-1,0)
(0,-1)
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Tell me what you know about this circle.
(0,1)
(1,0)
(-1,0)
(0,-1)
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Typical responses include

• Its round.
• It has no corners.
• It has a diameter.
• It has a radius.
• It has .
• The area is
• The circumference is

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Lets look at the degrees.

• Degrees measure angles. What are some angles we
  can fill in to our circle?
• Halfway around the circle is a straight angle or
  
• A quarter of the way around is a right angle or
• Three-fourths of the way around the circle is

(0,1)
(1,0)
(-1,0)
(0,-1)
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We can further divide up our circle into smaller
sections. If we divide the first quadrant in
half, our angle is We can repeat this for each
of the remaining quadrants.
(0,1)
(-1,0)
(1,0)
(0,-1)
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I could have divided the 1st quadrant into
thirds.If so, the angles would be multiples of
30.This means my circle would look like this
(0,1)
(1,0)
(-1,0)
(0,-1)
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Joining the quarters and thirds would give us the
following circle
(0,1)
(1,0)
(-1,0)
(0,-1)
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This circle is called the unit circle because
the radius is 1 unit.Each angle is considered
to be in standard position because it starts at 0
degrees and rotates counterclockwise to the
terminal point which is where the leg of the
angle intersects the unit circle.Our next task
is to find the terminal point (x,y) for each
angle on the unit circle. We can use
properties of symmetry (x-axis, y-axis, and
origin) to help us complete this task very
quickly.
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Lets review our special triangles from geometry.

• In a 30-60-90 triangle with hypotenuse c, short
  leg a and long leg b
• c a x 2, so
• b a x , so

• In a 45-45-90 triangle with hypotenuse c and
  legs a
• c a x , so

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Using the special triangle relationship with t
45 and c 1
So, cos
And, sin
(0,1)
Because the angles are equal, x and y are equal,
so the sin ratio will be the same as the cos.
Y
t
(1,0)
(-1,0)
X
(0,-1)
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Interchanging the position of the 30 and 60
degree angles will switch the shorter leg to x
and the longer leg to y, so the sin and cos
values will trade.
(0,1)
(1,0)
(-1,0)
(0,-1)
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Terminal Point Coordinates
t
1 0
0 1
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To complete coordinates in the other quadrants,
use symmetry.

• In the second quadrant, points are symmetric
  across the y-axis so the coordinates will be
  (-x,y).
• In the third quadrant, points are symmetric
  across the origin so the coordinates will be
  (-x,-y).
• In the fourth quadrant, points are symmetric
  about the x-axis so the points will be (x,-y).

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The coordinates for each terminal point are as
follows
(0,1)
(1,0)
(-1,0)
(0,-1)
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• From geometry, we know sin(t), cos(t), and
  tan(t).
• Sin(t) is the ratio of the opposite side of the
  triangle to the hypotenuse.
• Cos(t) is the ratio of the adjacent side to the
  hypotenuse.
• Tan(t) is the ratio of the opposite side to the
  adjacent side.
• SOHCAHTOA!!!!

hypotenuse
opposite
t
adjacent
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By learning the unit circle coordinate values, a
variety of problems can be easily solved without
the use of a calculator.
For example Using the information shown, solve
the right triangle.
B
c
a6
C
b
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A handy tool for remembering the values of the
coordinates for the x or cos values and y or sin
values on the unit circle is the hand trick.

• Take your labels and write 0, 30, 45, 60, and 90
  on them.
• Place them on the fingers of your left hand (palm
  up)
• as follows
• Thumb 90
• Pointer 60
• Middle 45
• Ring 30
• Pinky 0

On your post-it note, write
and place it
on your
palm.

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Your hand should look like this






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• Here is how it works.
• Example Find cos .
• Fold down the finger with 60 on it.
• Count the number of fingers above the folded one.
• Put this number inside the radical on your
  post-it.
• This is the value of cos .
• You should have gotten .
• To find the sin , simply count the
  fingers below the folded one and place the number
  in the radical. The value is

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Now for the fancy stuff. What if you wanted to
know tan ? Knowing that tan(t)
, place your sin answer
over your cos answer and you will
get,
So, you can just put your radical sin number over
your radical cos number and you have tan. The
2s in the denominators will always cancel out!
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What about sec? Sec is the reciprocal function
of cos. Find the cos value and flip it, you now
have sec. This means you would use and
count the fingers above the folded one. For
csc, use the reciprocal of sin or and
count the fingers below the folded one. For
cot, use the reciprocal of tan or and
put the number above the folded one in the top
radical and the numbers below the folded one in
the bottom radical.
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When you get comfortable with it, you can use
the hand trick backwards when solving
trigonometric equations. Example
Can you figure out on your hand how to get an
answer of the whole number 3? Or, the whole
number 1?
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There is not a way to get the whole number 3.
This means that there is no value of x such that
sinx-3. In order to get an answer of 1, fold
down the thumb and you have 4 fingers below the
folded one. This would be . The
value of x where sin equals 1 is . While
this only works for the exact values on the unit
circle, it is really a time saver. Students
learn the first quadrant, use properties of
symmetry, and now they can figure out any exact
value problem for any trig function.
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Summary of Hand Trick
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My students really enjoyed learning this last
year. They found it much easier to remember
their exact values.

• If you would like copies of the slides, you may
  e-mail me at
• hayesv_at_brevard.k12.fl.us
• For a copy of the presentation, send a CD through
  the courier.