introduction to the unit circle and right triangle trigonometry presented by, ginny hayes space...
TRANSCRIPT
Introduction to the Unit Circle and
Right Triangle Trigonometry
Presented by,
Ginny Hayes
Space Coast Jr/Sr High
Draw the circle
Label the x- and y-intercepts. Your circle should look like this:
122 yx
(1,0)
(0,1)
(-1,0)
(0,-1)
Tell me what you know about this circle.
(1,0)
(0,1)
(-1,0)
(0,-1)
Typical responses include:
• It’s round.• It has no corners.• It has a diameter.• It has a radius.• It has .• The area is • The circumference is
360.2r
.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 • A quarter of the way around is a right angle or • Three-fourths of the way around the circle is
.180
.90
.270
(1,0)
(0,1)
(-1,0)
(0,-1)
.90
.180
.270
360
0
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.
(1,0)
(0,1)
(-1,0)
(0,-1)
.90
.180
.270
360
45135
225 315
45
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:
.60 and 30
60
30
(1,0)
(0,1)
(-1,0)
(0,-1)
.90
.180
.270
360
60
30
120
150
210
245300
330
0
Joining the quarters and thirds would give us the following circle:
(1,0)
(0,1)
(-1,0)
(0,-1)
.90
.180
.270
360
60
30
120
150
210
245300
330
45135
225315
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 = a x 2, so
• b = a x , so
2c
a
3
cc
b23
32
•In a 45-45-90 triangle with hypotenuse “c” and legs “a”:
• c = a x , so 2
2c
a
x2
2
2
145So, cos
And, sin y2
2
2
145
(1,0)
(0,1)
(-1,0)
(0,-1)
t
Y
X
2
2,
2
2Because the angles are equal, x and y are equal, so the sin ratio will be the same as the cos.
Using the special triangle relationship with t= 45 and c = 1:2
,c
yx
2
1)30sin(
2
3)30cos(
y
x
(1,0)
(0,1)
(-1,0)
(0,-1)
.90
.180
.270
360
60
30
0
2
1,
2
3
2
3,
2
1
2
360sin
2
160cos
2
130sin
2
330cos
y
x
y
x
For the 30 central angle triangle, the shorter leg (y) hyp2
12
.
The longer leg (x) 3
2hyp 3
21 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.
Terminal Point Coordinates
yt sin
xt cos
t
1 0
0 1
0 30 45 60 90
2
3
2
12
2
2
2
2
1
2
3
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:
(1,0)
(0,1)
(-1,0)
(0,-1)
.90
.180
.270
360
60
30
120
150
210
245300
330
45135
225315
0
2
1,
2
3
2
2,
2
2
2
3,
2
1
2
3,
2
1
2
2,
2
2
2
1,
2
3
2
1,
2
3
2
2,
2
2
2
3,
2
1
2
1,
2
3
2
2,
2
2
2
3,
2
1
• 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
adjacent
opposite
adj
oppt
hyp
adjt
hyp
oppt
)tan(
)cos(
)sin(
t
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 righttriangle.
a=6
30A
B
b
c
C
362
312
1230cos30cos
1262
1630sin
630sin
bbb
c
b
cccc
603090 B
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.2
90
60
45
30
0
2
Your hand should look like this:
Here is how it works.
Example: Find cos .
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 .
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
60
60
2
1
60
2
3
Now for the fancy stuff.
What if you wanted to know tan ?
Knowing that tan(t)= ,
place your sin answer over your cos answerand you will get,
60
)cos(
)sin(
t
t
x
y
adj
opp
60 60
31
3
2123
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 valueand 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 thenumber above the folded one in the top radical and thenumbers below the folded one in the bottom radical.
2
2
When you get comfortable with it, you can use thehand trick backwards when solving trigonometricequations.
Example:
1sin 3sin
01sin 03sin
01sin3sin
03sin2sin 2
xx
xx
xx
xx
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. Thismeans that there is no value of x such that sinx=-3.
In order to get an answer of 1, fold down the thumband 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 theunit circle, it is really a time saver. Students learnthe first quadrant, use properties of symmetry,and now they can figure out any exact value problemfor any trig function.
12
2
2
4
90
Summary of Hand Trick
foldedabovefingersx
#
2)sec(
2
#)cos(
foldedabovefingersx
foldedbelowfingersx
#
2)csc(
2
#)sin(
foldedbelowfingersx
foldedabovefingers
foldedbelowfingersx
#
#)tan(
foldedbelowfingers
foldedabovefingersx
#
#)cot(
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:
• For a copy of the presentation, send a CD through the courier.