Advanced Geometry

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Advanced Geometry

• 9.9 Introduction to Trigonometry

SIN
COS
TAN
2
Sign? Co sign?
Tangerine?
3
How is trigonometry useful?
Suppose you wanted to know the height of a VERY
tall building . . .
4
OR, . . . maybe somebody (desperately) needs to
know the height of a REALLY TALL tree!
So, that means Im only about 15 feet from at
least 32 feet of safety!
h
15 more feet!
65
5
Common Triple?
Pythagorean Theorem?
Family?
Special Right ? ?
Trigonometry helps you solve for missing
measures in right triangles when you dont have
all the information you need in order to use the
methods youve already learned!
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?ABC ?ADE ?AFG
F
Lets review
D
B
x
1
2
3
A
C
E
G
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?ABC ?ADE ?AFG
F
All three triangles are similar by AA.
D
B
x
1
2
3
A
C
E
G
?A is one acute corresponding angle in each of
the three triangles
(reflexive property)
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?ABC ?ADE ?AFG
F
All three triangles are similar by AA.
D
B
x
1
2
3
A
C
E
G
Angles 1, 2, and 3 are right angles,
. . . and ALL right angles are congruent.
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?ABC ?ADE ?AFG
All three triangles are similar by AA.
F
The third angle in each triangle has no choice,
since the sum of the angles in any triangle is
exactly 180!
D
B
x
1
2
3
A
C
E
G
(That is why AA is sufficient to prove the three
triangles similar.)
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?ABC ?ADE ?AFG
All three triangles are similar by AA.
B
1
x
A
C
Since ?ABC, ?ADE, and ?AFG are similar triangles,
the following ratios are equal
That is because they represent corresponding
sides of the three similar triangles.
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?ABC ?ADE ?AFG
All three triangles are similar by AA.
90 - x
B
90 - x
90 - x
1
x
A
C
NOTE No matter what the measure of ?A, the
other acute angles equal 90 (m ?A).
Remember, the acute angles in each right triangle
have a complementary relationship!
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?ABC ?ADE ?AFG
90 - 30
B
90 - 30
90 - 30
30
A
C
Before we go on, it may help to review some
concepts you learned in a previous section about
one of the special triangles, the 30 60 90
right triangle.
Remember in the 30-60-90 Right Triangle, there is
a certain relationship that the sides always
obey. That relationship can be expressed using
the following ratio,
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?ABC ?ADE ?AFG
90 - 30
60
g
B
90 - 30
e
60
a
c
a
90 - 30
a
60
30
A
C
b
d
f
a b c a d e a f g 1
v3 2
The short leg (usually a) is opposite the
smallest (30) angle,
The long leg (usually b) is opposite medium (60)
angle, and
The hypotenuse (usually c) is ALWAYS opposite the
right (90) angle,
which is the largest angle. . . so it is the
LONGEST side!
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?ABC ?ADE ?AFG
Proper Notation
VERTICES are labeled using capital letters
SIDES are labeled using lower case letters
f
60
B
d
60
b
60
e
g
a
a
a
c
30
A
C
a d e
a f g
a b c
The short leg (a) is opposite the smallest (30)
angle,
The long leg (b d f ) is opposite medium
(60) angle, and
The hypotenuse (c e g) is ALWAYS opposite the
right (90)angle, which is the largest angle . .
. so the longest side
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?ABC ?ADE ?AFG
60
B
g
e
a
60
a
c
a
60
30
d
A
C
b
f
a d e
a f g
a b c
30 60 90
small medium large
1 v3 2
x xv3 2x
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Using the figure from the beginning, this time we
will let m? A 30, BC 1, DE 2, and FG x
F
D
x
B
2
1
3
30
A
C
E
G
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Taken apart, and according to the ratio of sides,
along with these measures we have
60
B
2x
60
4
x
2
2
60
1
30
A
C
In ? ABC In ?
ADE In ? AFG
BC 1 DE
2 1 FG x
1

AB 2
AD 4 2
AF 2x 2
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Taken apart, and according to the ratio of sides,
we have
60
B
2x
60
4
x
2
2
60
1
30
A
C
In each of the triangles above, the comparison,
or ratio, of the lengths of the sides of the
angle across from the 30 reference angle is 1
2 or 0.5!
Opposite LEG to HYPOTENUSE 1 2 or 0.5!
This ratio will be true for every 30 60 90
right triangle, no matter how small or large !
If 30, then opposite side to hypotenuse
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Now for the interesting part!
What do you think would happen if we changed ? A
to another acute measure?
Do you think the ratio of the length of the side
located opposite of the angle to the length of
the hypotenuse will remain constant ?
The answer is YES!
opposite
Question Why?
hypotenuse
Well . . . what have you learned that proves
figures similar?
57
74
63
Answer
A
The ANGLES are CONGRUENT, and CORRESPONDING
SIDES of SIMILAR figures are PROPORTIONAL!
In fact, in right triangles, given an ACUTE angle
of x degrees, the length of the side opposite
the angle always compares to the length of the
hypotenuse by the same ratio !
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Just in case you are not convinced, lets try it
with the other special right triangle, the 45
45- 90.
x
2
1
45
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Taken apart, and according to the ratio of sides,
we have
45
45
xv2
x
B
2v2
2
45
v2
1
45
A
C
In ? ABC In ?
ADE In ? AFG
BC 1 DE
2 1 FG x
1

AB v2
AD 2v2 v2
AF v2x v2
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Sure enough, in each similar triangle from
before, the ratio of the leg opposite the
reference angle does compare to the hypotenuse
length in the same way
In the 30 60 90 triangle, the ratio of the
leg opposite the 30 angle to the hypotenuse
1 2. And,
in the 45 45 90 triangle, the ratio of the
leg opposite the 45 angle to the hypotenuse
1 v2 (In simplest radical form, v2 )


2
Notice that in these examples, we have been
comparing the side located opposite of the 30 or
45 degree angle to the hypotenuse of these
triangles.
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For any given acute angle, the ratio of the sides
opposite of the angle will be the SAME,
regardless of the size of the RIGHT triangle.
From now on we will refer to this type of ratio
by its special name THE SINE RATIO.
The SINE RATIO of an acute angle of a right
triangle is the comparison of the length of the
side opposite the reference angle to the length
of the hypotenuse.
hypotenuse
Opposite to Hypotenuse
Opposite side
Reference Angle
Opposite Hypotenuse
Opposite Hypotenuse
SINE Ratio
SIN
Calculator SIN
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Either ACUTE ANGLE can be the reference angle,
but NEVER the RIGHT angle!
Okay, so now we will tie up some loose ends!
Reference Angle The acute angle in the right
triangle from which you view the other parts of
the given triangle.
Opposite Side
(LEG)
Adjacent Side
(LEG)
Hypotenuse
Opposite Side
Hypotenuse
Reference Angle
Adjacent Side
Opposite across from
Adjacent next to
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So, working from the other acute angle, we have
A reversed perspective!
Reference Angle The acute angle in the right
triangle from which you view the other parts of
the given triangle.
Reference Angle
Opposite Side
(LEG)
Adjacent Side
(LEG)
Hypotenuse
Adjacent Side
Hypotenuse
Opposite Side
Opposite across from
Adjacent next to
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In the earlier examples, we were exploring one of
the three trigonometric ratios introduced in this
section
The SINE ratio
So again, the sine ratio of an acute angle in any
right triangle is the value you get when the
length of the side opposite the reference angle
is divided by the length of the hypotenuse.
Opposite Leg
? SINE ratio
Hypotenuse
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In a right triangle, other ratios may also be
formed, and each has a special name
SINE, COSINE, and TANGENT
(SohCahToa)
SINE
COSINE
TANGENT
Opposite Leg
Adjacent Leg
Opposite Leg
Hypotenuse
Hypotenuse
Adjacent Leg
Soh
Cah
Toa
Soh
Cah
Toa
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SohCahToa is a memory device that helps you
remember which sides of the right triangle you
need for each trigonometric ratio, and in what
order they should be entered into the ratio!
(SohCahToa)
tan ?A
sin ?A
cos ?A
SINE
COSINE
TANGENT
Opposite Leg
Adjacent Leg
Opposite Leg
Hypotenuse
Hypotenuse
Adjacent Leg
Soh
Cah
Toa
Soh
Cah
Toa
Think!
Think!
Think!
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Definition A trigonometric ratio is the
comparison of the lengths of any two sides of a
RIGHT triangle when taken from the perspective of
one of the two acute angles.
Given Right ? ABC, with legs 3 and 4, and
hypotenuse 5, find
A
0.6
Soh
1) sin ?A
0.8
Cah
5
2) cos ?A
4
Hypotenuse
leg
0.75
Toa
3) tan ?A
B
leg
C
3
Remember, every ratio can be written as a
decimal!
See the table on PAGE 424 in your book for ratios
related to acute angles whose measures are
between 0 and 90 degrees!
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What is the result if we use ?B as our reference
angle?
Given Right ? ABC, with legs 3 and 4, and
hypotenuse 5, find
A
0.8
Soh
1) sin ?B
0.6
Cah
5
2) cos ?B
4
Hypotenuse
leg
1.333
Toa
3) tan ?B
B
leg
C
3
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Next lets compare the ratios written from the
two different reference angles, ?A and ?B
Given Right ? ABC, with legs 3 and 4, and
hypotenuse 5, find
A
Soh
sin ?A
sin ?B
Cah
5
cos ?A
4
cos ?B
Hypotenuse
leg
Toa
tan ?A
tan ?B
B
leg
C
3
1.333
0.8
0.75
0.8
0.6
0.6
The measure of the angles dont change!
It is simply the opposite perspective!
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Question Is it possible to create an
equilateral triangle using two 3-4-5 right
triangles?
Nope!
5
5
4
3
6
So, it is impossible for the acute angles in a 3
4 5 right triangle, and those in the same
family, to have measures of 30 and 60 degrees!
RIGHT?
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So, to find the measure of the acute angles we
can use the ratios from before
( 0.75)
Toa
A
tan ?CAB
0.75
37
5
tan ?CAB
4
?CAB (tan-1) (0.75)
?CAB 37
B
3
C
Negative Exponent NOTE tan-1
means you are dividing 0.75 by tan.
Now, if m?CAB 37 , then the m?B 90 37
53 . . . RIGHT?
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Check it by using the tangent ratio written from
?B!
( 1.333. . .)
Toa
A
tan ?B
1.333. . .
5
tan ?B
4
?B (tan-1) (1.333. . .)
53
?B 53
B
3
C
Negative Exponent NOTE tan-1
means you are dividing 1.333. . . by tan.
Now, if m?B 53 , then the m?CAB 90 53
37 . . . Check!
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We will now verify the results achieved from the
tangent ratio by calculating the angles with the
sine and cosine ratios.
VERY IMPORTANT! This means we could have
calculated the angle measures using any of the
three trigonometric ratios!
A
v
v
sin ?A
0.6
?A (sin-1) (0.6)
?A 37
v
37
5
4
v
?B (sin-1) (0.8)
?B 53
53
cos ?A
0.8
B
C
v
?A (cos-1) (0.8)
?A 37
3
v
v
Sine, Cosine or Tangent? The one you use depends
on the measures you know!
cos ?B
0.6
v
?B (cos-1) (0.6)
?B 53
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Example ?ABC is isosceles, with legs AB and AC.
Find sin ?C.
A
Step 1 First we must have a RIGHT ? !

If isosceles, then the altitude is also a MEDIAN!
Draw an altitude from ?A
Step 2 Use families of right triangles or the
Pythagorean Theorem to find the length of the
altitude.
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WHY? Remember SohCahToa ! The
sine ratio is the opposite leg
divided by the hypotenuse
Step 3 Write the ratio in fraction form,
opposite leg hypotenuse
B
C
14
7
7
sin ?C
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Example Find m ?C.
Step 1 Use the ratio you wrote for sin ?C.
A
Step 2 Write the ratio as a decimal
0.96
sin ?C
32
Step 3 Solve for ?C sin ?C 0.96
25
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sin ?C 0.96
?C sin-1 (0.96 )
74
74
B
C
m?C 74
7
7
Question What is the measure of ?A?
m?A 180 2( 74)
180 148
32
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EXAMPLE What if all you know is the measure of
one of the acute angles and
only one side length of the RIGHT triangle?
Opposite and Adjacent. . . Toa! Use the TANGENT
Ratio!
Problem Find the height of the flag pole.
Step 1 From the angle you know, determine
which trigonometric ratio to write.
We know Reference Angle 50
Adjacent Side 27 ft
We want to know Opposite Side
32 ft
h ft
Step 2 Write equation from the reference angle
Tan 50
1.1918
Tan 50 1.1918
50
27(1.1918) h
32 ft h
27 ft
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Calculator Did you notice?
When solving for a missing side,
You use these keys
These keys convert the acute angle measure to
its corresponding side ratio value, in decimal
form.
SIN
COS
TAN
When solving for an unknown angle measure
The inverse keys divide the decimal value of the
side ratio and render the corresponding angle
measure.
You use these keys
TAN-1
SIN-1
COS-1
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Trigonometry! Its almost as easy as A, B, C!
Student response to exam question 3
But it will require a bit more effort than this
student thought!
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9.9 Assignment
Pp 420 (2 4 9 14 18)