Download Introduction to the Unit Circle and Right Triangle

Introduction to the Unit Circle
and
Right Triangle Trigonometry
Presented by,
Ginny Hayes
Space Coast Jr/Sr High
Draw the circle
x  y 1
2
2
Label the x- and y-intercepts. Your circle
should look like this:
(0,1)
(1,0)
(-1,0)
(0,-1)
Tell me what you know about
this circle.
(0,1)
(1,0)
(-1,0)
(0,-1)
Typical responses include:
•
•
•
•
•
•
•
It’s round.
It has no corners.
It has a diameter.
It has a radius.
It has 360 .
2

r
.
The area is
The circumference is 2r.
Let’s 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 180 .

• A quarter of the way around is a right angle or 90 .

• Three-fourths of the way around the circle is 270 .
(0,1)
90 .
0 (1,0)
(-1,0)
360
180 .
(0,-1)
270 .
We can further divide up our circle into smaller sections.

45
If we divide the first quadrant in half, our angle is
We can repeat this for each of the remaining quadrants.
(0,1)
90 .
135
45
0
(-1,0)

360
180 .
225
315
(0,-1) 270 .

(1,0)
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) 90 .

120
30 and 60 .


60
150
30 
30
0
(-1,0)
180 .
360
210
330
300

60
240
(0,-1)
270.
(1,0)

Joining the quarters and thirds
would give us the following circle:

(0,1) 90 .

120

135
60
45
150
30
0
(-1,0) 180 .
360
210
225
330
315
240

(0,-1) 270 .
300
(1,0)

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.
Let’s review our special triangles from
geometry.
• In a 30-60-90 triangle
with hypotenuse “c”,
short leg = a and
long leg = b:
c
• c = a x 2, so a 
2
• b = a x 3 , so
c
3
b  3 c
2
2
•In a 45-45-90 triangle
with hypotenuse “c”
and legs “a”:
• c = a x 2 , so
c
a
2
c
x, y 
2

Using the special triangle relationship with t= 45 and c = 1:
1

So, cos 45  
2

1

And, sin 45  
2

2
x
2
2
 y
2
(0,1)
Because the
angles are
equal, x and y
are equal, so the
sin ratio will be
the same as the
cos.
 2 2


 2 , 2 


Y
t
(-1,0)
(1,0)
X
(0,-1)
hyp
For the 30 central angle triangle, the shorter leg (y)  2  12 .
The longer leg (x) 
3
2
hyp 
3
2

1
3
2
.
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.

x  cos 30

y  sin 30








x  cos 60  
y  sin 60  
3
2
1
2
1
2
3
2

(0,1) 90 .
3
2
1
y  sin( 30 ) 
2
x  cos(30 ) 
(-1,0) 180 .
1
3
 ,



2
2
60 

30

 3 1


 2 ,2


0
360
(0,-1)
270.
(1,0)

Terminal Point Coordinates
t
cos t  x
sin t  y



0 30 45
1
0
3
2
1
2

60 90
2
2
1
2
0
2
2
3
2
1

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).
The coordinates for each terminal point are
as follows:
 1
3
 ,

 2 2 


 2 2


 2 , 2 




(0,1) 90 .
120
1
3
 ,

2 2 
60   2  2 

45 
135

150
1
2
30

3


 2 ,2



2 
,
 3 1


 2 ,2


0
(-1,0) 180 .

3 1


 2 , 2 


360
 3 1

, 
 2
2
330 
210

2  225


2

 2 , 2 


 1
3
  ,

 2
2 


(1,0)

315 
240

(0,-1) 270 .
2
2


 2 , 2 



1
3
 ,

2
2 


300
hypotenuse
opposite
t
adjacent
• 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!!!!
opp
sin( t ) 
hyp
adj
cos(t ) 
hyp
opp
tan( t ) 
adj
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.
6
1

sin 30   c sin 30  6  c   6  c  12
c
2

B
c
a=6
 3
b
b

bb6 3
cos 30   cos 30   12

c
12
2



B  90  30  60
C
b
A  30 
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
On your post-it note, write
•Pointer:
60
and
•Middle:
45
place it
•Ring:
30
on your
•Pinky:
0
palm.
2
Your hand should look like this:
90

60
45
2
30
0

Here is how it works.
Example:
Find cos 60  .

1. Fold down the finger with 60 on it.
2. Count the number of fingers above the folded one.
3. Put this number inside the radical on your post-it.
4. This is the value of cos 60  .
 
You should have gotten
1
2
.
To find the sin 60 , simply count the fingers

below the folded one and place the number in the
radical. The value is 3
2
Now for the fancy stuff.
What if you wanted to know tan
Knowing that tan(t)=
opp y sin( t )
 
adj x cos(t )
60 ?

,
place your sin 60  answer over your cos 60  answer

and you will get,

3
2  3 
1
1
2
3
So, you can just put your radical sin number over your
radical cos number and you have tan. The 2’s in the
denominators will always cancel out!
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
2
and count the fingers above the folded one.
2
For csc, use the reciprocal of sin or
fingers below the folded one.
and count the
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.
When you get comfortable with it, you can use the
hand trick backwards when solving trigonometric
equations.
Example:
sin x  2 sin x  3  0
2
sin x  3sin x  1  0
sin x  3  0 sin x  1  0
sin x  3
sin x  1
Can you figure out on your hand how to get an
answer of the whole number 3? Or, the whole
number 1?
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 24  22  1 . The value of x where sin equals

1 is 90 .
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.
Summary of Hand Trick
# fingers above folded
cos( x) 
2
sec( x) 
2
# fingers above folded
sin( x) 
# fingers below folded
2
csc( x) 
2
# fingers below folded
tan( x) 
# fingers below folded
cot( x) 
# fingers above folded
# fingers above folded
# fingers below folded
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:
[email protected].
fl.us
• For a copy of the
presentation, send a
CD through the
courier.