Transcript TI84

Author: Kyle Heffelbower
Trigonometric functions- sine, cosine, and
tangent
 Sine is abbreviated sin
 Cosine is abbreviated cos
 Tangent is abbreviated tan
 The Greek letter theta (Θ) is used as a variable
for angles.
 Hypotenuse is the side opposite the right angle
in a right triangle. The longest side of a right
triangle.


For any RIGHT triangle the trigonometric
ratios allow us to find out information about
the side lengths and angle measures when
given some basic information

The sine of an angle for
a right triangle, is
equivalent to the ratio
of an opposite side and
hypotenuse
opposite
sin( ) 
hypotenuse
Hypotenuse
Opposite side
with respect
to theta
Θ
Adjacent side
with respect
to theta

The cosine of an angle
for a right triangle, is
equivalent to the ratio
of an adjacent side and
hypotenuse
adjacent
cos() 
hypotenuse
Hypotenuse
Opposite side
with respect
to theta
Θ
Adjacent side
with respect
to theta

The tangent of an
angle for a right
triangle, is equivalent
to the ratio of an
opposite side and the
adjacent side
opposite
tan( ) 
adjacent
Hypotenuse
Opposite side
with respect
to theta
Θ
Adjacent side
with respect
to theta

All of these relationships can easily be
remembered with the acronym
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t

Now that we have done a bit of review, let’s
get some practice in before we get to the new
stuff.

Choose the best answer for the length of the
h given the right triangle below.
13 meters
a. 9.959 meters correct
b. 8.356 meters cos
c. -3.411 meters rad
d. 16.970 meters divide
50°
h



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.

Nice job recognizing that it is a right triangle
and you need to use sine to complete it, but it
is the opposite side divided by the
hypotenuse- not the other way around.
Let’s take another look


Let’s take another look at the trigonometric
ratios and try this out again.
Make sure you are identifying the sides
accurately.
Let’s try this again!


Way to go champion!!
Keep up the good work.

Choose the best answer for the length of the
x
h given the right triangle below.
38°
a. 10.152 meters divide
b. 6.304 meters correct
c. 4.925 meters sin
d. 7.641 meters rad
8 meters



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.

Nice job recognizing that it is a right triangle
and you need to use cosine to complete it,
but it is the adjacent side divided by the
hypotenuse- not the other way around.
Let’s take another look


Let’s take another look at the trigonometric
ratios and try this out again.
Make sure you are identifying the sides
accurately.
Let’s try this again!


Way to go champion!!
Keep up the good work.

Choose the best answer for the length of the
h given the right triangle below.
3 meters
24°
a. -6.405 meters
b. 6.738 meters
c. 1.220 meters
d. 1.336 meters
h



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.

Nice job recognizing that it is a right triangle
and you need to use tangent to complete it,
but it is the opposite side divided by the
adjacent- not the other way around.
Let’s take another look


Let’s take another look at the trigonometric
ratios and try this out again.
Make sure you are identifying the sides
accurately.
Let’s try this again!


Way to go champion!!
Keep up the good work.

Now that we are caught up to speed in the
review of the solving trigonometric ratios, it
is time to get to the graphing.

A ride maker decides to
crank up the intensity of
the “Megaloop.” She
wants to place half of it
underground. Always
conscious of safety and
evacuation necessities
she has called on you to
discover the distance
from the ground at any
point on this loop. See if
you can help her out.
Job one: Let’s get a picture of the
scenario.





Draw a circle to fit the
ride.
Next draw a rectangle
to model the
“underground” portion
of the ride.
Now, to get a sense of
this let’s create axes.
Last, let’s identify the
radius of this circle.
Nice drawing!!
y


1
x
Just for ease of
calculation let’s call the
radius 1 unit long.
Calling the radius 1 unit
long gives us a great
model of a common
mathematical model
named the
Unit Circle

(1 unit radius= unit circle)

As with our evacuation
route for the half
underground
Megaloop, the goal is
to find the distance
from the ground. In
the picture this is the
vertical distance from
the x-axis.
y
1
y
x

Time to calculate the
evacuation distance.
For the ride. If you
need help just call on
Pythagoras.
y
1
Θ
y
x



For the next activity you will need a scientific
calculator in degree mode.
We will take a look at several different right
triangles developed from various angles and
build a table of values.
Let’s get cracking!!
y

Find the height at 30
degrees.
 0.866 cos
1
30°
y
x
 0.577 tan
 0.5 correct
 -0.988 radians


Great work calculating that distance.
Let’s put that on the table and continue!!
Bring on the next one!!

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
Height
0.5
45
60
90
y
120
150
180

1
45°
y x
210
240
270
300
330
Find the height at 45
degrees.




0.851 radi
0.707 correc
1 tan
0.707 cos
360


Nice job putting that together.
Let’s put that on the table and continue!!
Bring on the next one!!

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
Height
0.5
0.707
60
90
y
120
150
180

1
60°
y
x
210
240
270
300
330
Find the height at 60
degrees.




0.5 cos
0.866 correc
-0.305 radi
1.732 tan
360


Distance is just right.
Let’s put that on the table and continue!!
Bring on the next one!!

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
Height
0.5
0.707
0.866
90
y
1
120
150
180

y
90°
x
210
240
270
300
330
Find the height at 90
degrees.




0.894 radi
1 correc
undefined
0 cos
360

Let’s put that on the table and continue!!
Bring on the next one!!

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
90
Height
0.5
0.707
0.866
1
y
y
120
150
180

1
120°
x
210
240
270
300
330
360
Find the height at 120
degrees.




0.581 radi
-0.5 cos
-1.732 tan
0.866 correct

Let’s put that on the table and continue!!
Bring on the next one!!

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
90
120
Height
0.5
0.707
0.866
1
0.866
y
y
150
180

1
150°
x
210
240
270
300
330
360
Find the height at 150
degrees.




-0.715 radi
-0.866 cos
-0.577 tan
0.5 correct


Are you beginning to sense a bit of a pattern?
Let’s put that on the table and continue!!
Bring on the next one!!

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
90
120
150
Height
0.5
0.707
0.866
1
0.866
0.5
y
180°
180

x
210
240
270
300
330
360
Find the height at 180
degrees.




1 cos
-0.801 radi
Undefined arithmeti
0 correct


Keep on trucking!!
Finish this up.
Bring on the next one!!

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
90
120
150
180
Height
0.5
0.707
0.866
1
0.866
0.5
0
y
210°
y
1

x
210
240
270
300
330
360
Find the height at 210
degrees.




-0.5 correct
-0.866 cos
0.577 tan
0.468 radin

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
90
120
150
180
210
Height
0.5
0.707
0.866
1
0.866
0.5
0
-0.5
y

240°
y
1
x
240
270
300
330
360
Find the height at 240
degrees.




1.732 tan
0.945 rad
-0.866 corredc
-0.5 cos

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
90
120
150
180
210
240
Height
0.5
0.707
0.866
1
0.866
0.5
0
-0.5
-0.866
y
270°
-1 y

x
270
300
330
360
Find the height at 270
degrees.




-0.176 radi
-1 correc
undefined
0 cos

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
90
120
150
180
210
240
270
Height
0.5
0.707
0.866
1
0.866
0.5
0
-0.5
-0.866
-1
y

x
300°
1
y
300
330
360
Find the height at 300
degrees.




0.5 cos
-0.866 correc
-1.000 radi
-1.732 tan

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
90
120
150
180
210
240
270
300
Height
0.5
0.707
0.866
1
0.866
0.5
0
-0.5
-0.866
-1
-0.866
y

x
330°
y
1
330
360
Find the height at 330
degrees.




-0.5 correct
0.866 cos
-0.577 tan
-0.132 radin

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Angle
30
45
60
90
120
150
180
210
240
270
300
330
Height
0.5
0.707
0.866
1
0.866
0.5
0
-0.5
-0.866
-1
-0.866
-0.5
y
360°

x
360
Find the height at 360
degrees.




1 cos
0.959 radi
Undefined arithmeti
0 correct

Remember our Trigonometric ratios and think
about what we were given.
S
i
n
e
O H
p y
p p
o o
s t
i e
t n
e u
s
e
C
o
s
i
n
e
A H
d y
j p
a o
c t
e e
n n
t u
s
e
T
a
n
g
e
n
t
O A
p d
p j
o a
s c
i e
t n
e t



While your idea is flawless and you did
correctly solve this, you did it for radians and
not degrees.
There are two popular ways to measure
angles- degrees is one, radians is the other
Whenever you are working with
trigonometric functions- sine, cosine,
tangent- you need to make sure your
calculator is in the right MODE
Help with switching you calculator
Try again



For the TI-81 through TI-84 the radian/degree
shift is in the “mode” menu. The button is
next to the 2nd key.
For the TI-nSpire the radian/degree shift is in
the general settings menu
Many scientific calculators have a units shift
button. On the screen it will say either “deg”
or “rad” in little letters.
Whew!! Now that we have created a great table
we need to graph it.
 But first…
 Look back at your calculations. Did you notice
any commonalities?

 Nope. I did not recognize nuttin’.
 Now that you mention it, I do notice that we did a lot
of sine of the angle.
 Uh- duh?! Of course! We did the same trigonometric
ratio every single time. In fact, I stopped writing it out
after the first four because I am awesome!


Look closely at each of the times you
calculated the height. Every single time you
were given an angle, the hypotenuse and you
needed to find the opposite side.
Angle… opposite… hypotenuse
opposite
sin  
hypotenuse
y
sin  
1
y  sin 
Let’s try that question again.


Good job recognizing the pattern!!
This means that ANYTIME you need to find
ANY height for ANY Megaloop evacuation
you will ALWAYS use sin(Θ)
Let’s get graphing this!!

Now that we have a table we can create the
graph.
Angle
30
45
60
90
120
150
180
210
240
270
300
330
360
Height
0.5
0.707
0.866
1
0.866
0.5
0
-0.5
-0.866
-1
-0.866
-0.5
0
1
0
-60
-30
0
-1
30
60
90
120
150
180
210
240
270
300
330
360
390
420




You have just created your first graph of a
trigonometric function.
This wave function you created is the graph of
f(x)= sin(x). This is the parent equation for the
sinusoidal family of functions.
We used the unit circle to generate this graph.
This means that for the Super Megaloop
designer who wants to place half of the ride
underground. She simply needs to remember
sin(x) to know all the evacuation heights!!



This wave function lends itself to many real
world uses.
Pendulums, light, sound, and most circle
motion can be graphed by these wave
functions.
Specific features and transformations will be
discussed later.
That led us to building right triangles
throughout which gave us a table of heights.
Angle
30
45
60
90
120
Height
0.5
0.707
0.866
1
0.866
And this built a unique family of functions
that look like waves.
1
0
-60-30 0 30 60 90120150180210240270300330360390420
-1
We noticed that while creating these heights
we ALWAYS simplified to…
The unit circle is a circle whose radius is
one unit and can be utilized to look at
trigonometric functions
y
y  sin( )
1
x
Restart from Beginning