Transcript Document

Chapter 7
Right Triangles and Trigonometry
7.1 Geometric Mean
Means
In the past, we have thought of Means as the
average. That is, the mean between 2 and 10 is
6 b/c (2+10)/2 = 6
You could notice that if you take 2 (starting
value) and add 4 you will get 6 (the mean).
If you take 6 (the mean) and add 4 again, you
get 10 (ending value)
Because you have added the same number
twice, we call this the Arithmetic Mean.
Today we will learn about Geometric Means.
Geometric Means
If for the Arithmetic Mean you add the same
number each time, what do you think you do to
get the geometric mean?
Right, you multiply by the same number each
time.
Let s be the starting number, e be the ending
number and g be the geometric mean.
So, sx = g and gx = e.
Solve both for x you get x=g/s and x=e/g
Continue
Since both equations equal the same thing,
you can combine them.
Notice that the geometric
e
g
means are written twice on one

g
s diagonal!
Notice that the starting number
and the ending number are on
the other diagonal!
The pattern will always be the same, the
GM’s are on one diagonal.
Find the GM between…
Find the Geometric Mean between 4 and
16.
Set up the equation 4/GM = GM/16
Cross Multiply and you get GM2 = 64
Taking the square root of each side you
get GM = + 8 but only +8 is between 4 and
16, then the GM between 4 and 16 is 8!
So, Why is Geometric Mean so important?
Geometric Means
C These three triangles are similar!
2
D
ΔABC ~ ΔADB ~ ΔBDC
4
1
A
3
B
AB BC AC


AD DB AB
Rearranging the proportions:
AC AB

AB AD
Notice AB is written twice on the diagonal?
So, AB is the GM between AD and AC!
Geometric Means Con’t
C
ΔABC ~ ΔADB ~ ΔBDC
D
AB BC AC


BD DC BC
A
B
Rearranging the proportions:
AC BC

BC DC
Notice BC is written twice on the diagonal?
So, BC is the GM between DC and AC!
Geometric Mean Con’t
C
ΔABC ~ ΔADB ~ ΔBDC
D
AD DB AB


BD DC BC
A
B
Rearranging the proportions:
AD DB

BD DC
Notice BD is written twice on the diagonal?
So, BD is the GM between DC and AD!
7.2 Pythagorean Theorem
and Converse
Pythagorean Theorem
Pythagoras recognized a very important
relationship between the sides (legs) and
the hypotenuse of a right triangle.
That is “The sum of the squares of the
legs equals the square of the hypotenuse.”
B
c
Here we have a2 + b2 = c2
a
I like leg2 + leg2 = Hypot2
A
b
C
Converse of Pythagorean
Theorem
Did you notice that the only triangle that
uses the Pythagorean theorem is a right
triangle?
So, if the “sum of the squares of the two
smaller sides equals the square of the
largest side” then the triangle is a Right
Triangle.
If leg2 + leg2 = Hypot2, then the triangle is
a right triangle.
Corollaries of Pythagoras
There are two corollaries of the
Pythagorean theorem that allows you to
classify a triangle by angles.
If a2 + b2 > c2 (where a and b are the two
smallest sides), then the triangle is acute.
If a2 + b2 < c2 (where a and b are the two
smallest sides), then the triangle is obtuse.
Pythagorean Triple
There are all sorts of side combinations
that we can use to make a right triangle
but some are more special.
When all three sides of a right triangle are
integers, then these three numbers are
called a “Pythagorean Triple”
Some common triples are 3, 4, 5 or 5, 12,
13, or 7, 24, 25…..
Plus all families of these… 3x, 4x, 5x etc..
7.3 Special Right Triangles
Special Right Triangles
Let’s take this equilateral
60°
triangle, all three sides and angles
are congruent and all three angles
60°
60° measure 60°
Now let us drop an altitude from the top of
the triangle to the horizontal side.
Remember that when an altitude is drawn
from the vertex angle of an “isosceles”
triangle it is also an angle bisector and a
median?
Special Right Triangle
(Con’t)
60°
60°
60°
So, what do we know about the
bottom side and what do we know
about what is happening at the top
angle?
Each angle is 30°
If the side is s then each piece is ½ s.
The bottom segment is divided in ½ and so is
the vertex angle.
30°-60°-90° Right Triangles
Long Leg
Short Leg (SL) – This is the leg opposite the
30° angle.
30°
Long Leg (LL) – This is the leg
opposite the 60° angle.
Hypotenuse (H) – This is the side
opposite the right angle.
60°
Short Leg
H → SL, H/ 2 = SL
Rules:
SL → H, SL x 2 = H
SL → LL, SL x √3 = LL
LL → SL, LL / √3 = SL
45°- 45°- 90° Right Triangles
Let us look at this square, it
is equilateral and equiangular.
Let us divide the square by
drawing a diagonal.
Now we have an “Isosceles, Right Triangle”
or a 45-45-90 Right Triangle.
If we make the legs 1, then what is the
length of the Hypotenuse? √2
45°- 45°- 90° Right Triangles
Rules Recap:
45 – 45 - 90
Leg to Hypotenuse: L x √2 = H
Hypotenuse to Leg: H ÷ √2 = L
30 – 60 – 90
SL to Hypotenuse: SL x 2 = H
Hypotenuse to SL: H ÷ 2 = SL
SL to LL: SL x √3 = LL
LL to SL: LL ÷ √3 = SL
Practice
b
B
SL
2
30°
c
LL
Hypot
6
a
c
a
3
3√3
A
b
5
C
2√2
2√2
Practice
b
B
a
c
A
b
C
c
a
SL
2
LL
2√3
Hypot
4
3
3
3
5/2
2√2
√2
3√3
3√3
3√3
5√3/2
2√6
√6
6
6
6
5
4√2
2√2
More Practice
C
b
a
45°
A
c
c
b
a
Leg
3
Leg
Hypot
B
4
6√2
7√2
8
3√2
4√3
More Practice
C
b
a
45°
A
c
B
c
b
a
Leg
3
4
6
7
4√2
3√2
4√3
Leg
3
4
6
7
4√2
3√2
4√3
Hypot
3√2
4√2
6√2
7√2
8
6
4√6
7.4 Trigonometry
Why Trig?
Up to now we have been able to find sides and
or angles of certain types of Right Triangles.
If we knew two sides, we could use the
Pythagorean Theorem to find the third side.
If we saw it followed a pattern, say a
Pythagorean Triple, we could find missing sides.
If it was one of the two special right triangles, we
could apply the rules.
What happens if we can’t fit any of these
situations?
Why Trig?
We can use Right Triangle trig if we have
a right triangle and a known acute angle
and side to find all the missing sides.
We can use Right Triangle trig if we have
a right triangle and we know two sides to
find any missing acute angle.
Trig Ratios
There are three important trig ratios we
will use in geometry (there actually are six
– but that will have to wait 2 years).
They are Sine (Sin), Cosine (Cos) and
Tangent (Tan).
You can see the Sin, Cos and Tan buttons
on your calculator.
Before we do anything, make sure you are
in Degree Mode vice Radian Mode.
Trig Ratios (Con’t)
The sine of an acute angle is the ratio of the
opposite side over the hypotenuse.
The cosine of an acute angle is the ratio of
the adjacent side over the hypotenuse.
The tangent of an acute angle is the ratio of
the opposite side over the adjacent side.
SOH/CAH/TOA
Some Old Hippie, Caught Another Hippie,
Tripp’n Over Animals.
Sine and Cosine
C
b
A
c
a
B
Sin <A = Opposite Side
Hypotenuse
= a
b
Sin <C = Opposite Side
Hypotenuse
Cos <A = Adjacent Side
Hypotenuse
= c
b
= c
b
cos <C = Adjacent Side
Hypotenuse
= a
b
Tangent
C
b
A
c
a
B
Tan <A = Opposite Side
Adjacent Side
= a
c
Tan <C = Opposite Side
Adjacent Side
= c
a
To find the missing sides of a right
triangle, you must use either the Sine,
Cosine or Tangent functions Depending
on what they give you in the problem.
Calculator
Turn your calculator on… got to Mode,
make sure you’re on Degree Mode.
To find the sine of 39° all you need to do is
type “sin 39” and you’ll get …
sin 39° = .629320391
What does that mean?
That is the ratio of the length of the
opposite side of the 39° angle over the
length of the hypotenuse.
Calculator (Con’t)
Find the cosine of 47°
cos 47° = .6819983601
That is the ratio of the adjacent side over
the hypotenuse is .6819983601
Find the tangent of 21°
tan 21° = .383864035
That is the ratio of the opposite side over
the adjacent side.
Example
C
b = 20
a
What do we know?
25°
A
Given this triangle find a.
c
B
We know we have a right triangle, an
acute angle that measures 25° and the
length of the hypotenuse.
We want to find a That is the opposite side,
So which function?
Example (Con’t)
C
b = 20
a
25°
A
c
B
We have or want the
opposite side and the
hypotenuse… what function
has Opp Side and Hypot?
Sine function…. So sin 25° = a/20.
Solving for a …. a = 20 sin 25° or approx 8.45
Now lets find c …. cos 25° = c/20
Solving for c …. c = 20 cos 25° or approx 18.13
Another Example
C
b
Given this triangle find a.
a
What do we know?
35°
C = 20
B
We know we have a right triangle, an
acute angle that measures 35° and the
length of the adjacent side from <A.
We want to find a That is the opposite side,
So which function?
Example (Con’t)
C
b
a
35°
A
c = 20
B
We have or want the
opposite side and the
adjacent side… what function
has Opp Side and Adj Side?
Tangent function…. So tan 35° = a/20.
Solving for a …. a = 20 tan 35° or approx 14.00
Now lets find b …. cos 35° = 20/b
Solving for c …. b = 20/cos 35° or approx 24.42
Angles?
So, all the examples we have done so far
have had you find sides. You used sine,
cosine and tangent functions.
To find angles, it is pretty similar except
you will use Inverse Sine, Inverse Cosine
and Inverse Tangent.
Inverse Sine is sin -1.
Inverse Cosine is cos -1.
Inverse Tangent is tan -1.
Angles (Con’t)
So to find the angle that has a sine of
.8543 all you need to do is type
sin -1 (.8543) and you’ll get the angle
measurement.
sin -1 (.8543) = 58.7°
Example
C
b
A
c = 20
a = 15
Find the measure of <A
What do we know?
B
We know from Angle A , that a = 15 and
c = 20.
a is the opposite side, c is the adjacent side
from <A so we’re going to use Inv Tan
So, tan -1 (15/20) = 36.9°
7.5 Angles of Elevation and
Depression
Angle of Elevation
How do you go about finding
the height of a building that a
person is standing on?
You can use right triangle trig.
So you need to draw a right triangle.
Angle of Elevation and
Depression
Horizontal
Angle of Depression
Angle of Elevation
Horizontal
< of Depression
Elevation is
measured
is
measured
from
the horizontal
from
the
up
to the “linedown
horizontal
of
sight.”
to
the “line of
sight.”
Angle of Elevation and
Depression
The measurements of the angle of
elevation and depression are the same
because they are….
Alternate Interior Angles made by a
transversal “line of sight” cutting parallel
lines “horizontal lines.”
Do not think that the complementary angle
of the angle of elevation is the angle of
depression.
Example
Find the height of a cliff if you are in a sail
boat 2000’ from the cliff and the angle of
elevation to the top is 13°
y
13°
2000’
y
tan13 
2000
y  2000 tan13
y  461.74 '
7.6 Law of Sines (H)
What if?
What if you don’t have a right triangle?
Then depending on the information you
can use either the Law of Sines or the Law
of Cosines.
We will only cover the Law of Sines in this
class, we will leave the Law of Cosines for
Pre Calculus.
Law of Sines
C
b
a
h
A
c
D
ΔACD
ΔBCD
sin A = h/b
sin B = h/a
h = b sin A
h = a sin B
b sin A = a sin B
B
Law of Sines
sin A sin B sin C


a
b
c
C
b = 10
a
35°
A
45°
c
B
Example
A
10
C
100° a
35°
45°
c
10sin35  a sin 45
10 sin 35
a
sin 45
a  8.11
sin A sin B sin C


a
b
c
B
sin 35 sin 45 sin100


a
10
c
c sin 45  10sin100
10sin100
c
sin 45
c  13.93
Unit Circle and Radians
Unit Circle
Arc Length
=1
Unit Circle – a circle
w/ a radius = 1.
r=1
< meas
= 1.
A radian – an angle
measurement that
gives an arc length
= to the radius.
Unit Circle
90° or π/2
120° or 2π/3
60° or π/3
45° or π/4
135° or 3π/4
30° or π/6
150° or 5π/6
180° or π
0°/360° or 2π
330° or 11π/6
210° or 7π/6
315° or 7π/4
225° or 5π/4
240° or 4π/3
300° or 5π/3
270° or 3π/2
Radians
Radians is a unit that we use to measure angles.
It is different unit of measure than degrees. Just
like cm’s measure length and so do inches.
Conversion factor πr = 180°
The “r” is to show you that this is a radian
measure (not the ratio of Circumference divided
by the diameter)
Convert degrees to radians mult by (π/180°)
Convert radians to degrees mult by (180°/π)
Examples:
Given 3π/4 convert to degrees.
Take (3π/4)(180°/π)
You get 135°
Given 225° convert to π form radians.
Take (225°)(π/180°)
You get 5π/4