Download Unit08 PowerPoint for trigonometry class

Welcome
to Week 8
College
Trigonometry
Conics
Any plane cutting a cone,
regardless of its angle, will
produce a “conic”
Conics
Conics treasure hunt!
Conics
Circle: set of points in a plane
whose distances from a fixed
point (the center) are constant
(denoted "r")
( x - h ) 2 + ( y - k ) 2 = r2
Radius: r
Center: (h,k)
Conics
Ellipse: “squashed circle”
set of points in a plane whose
distances from two fixed points
(the foci) have a constant sum
Conics
Two types of ellipses:
horizontal
vertical
Conics
Horizontal ellipse:
(x - h)2 + (y - k)2 = 1
a2
b2
Vertical ellipse:
(y - k) 2 + (x - h)2 = 1
a2
b2
“a” is the long axis, “b” the short
one, (h,k) is the center
Conics
A circle is a special case of the
ellipse where the foci are the
same point
Conics
IN-CLASS PROBLEMS
An ellipse has
Foci: (0, -4), (0, 4)
Vertices: (0, -6), (0, 6)
Is it horizontal or vertical?
Conics
Parabola: set of points in a
plane that are equidistant from
a fixed point F (the focus)
and a fixed line (the directrix)
ax2 + bx + c = y
Conics
Two types of parabolas:
N/S
y = a(x - h)2 + k
E/W
x = a(y - k)2 + h
Conics
Parabolas have a property that
makes them useful in the design
of reflectors and transmitters
Conics
A particle approaching a
parabola on a line parallel to
the axis is reflected through
the focus
Conics
This property is used to focus
incoming light by a parabolic
mirror on a telescope
Conics
Also, signals emanating from
the focus are reflected parallel
to the axis, a property used in
radio transmitters
and headlights
Conics
IN-CLASS PROBLEMS
Where would you place the
transmitter on a parabolic dish
antenna to maximize the
transmitted signal?
a)
b)
c)
d)
At
At
At
At
the
the
the
the
vertex
focus
directrix
outer edge of the dish
Conics
Hyperbola: set of points in a
plane whose distances from two
fixed points
(the foci) have
a constant
difference
Conics
Two types of hyperbolas:
E/W:
(x - h)2 - (y - k)2 = ±1
a2
b2
N/S:
(y - k)2 - (x - h)2 = ±1
a2
b2
Conics
IN-CLASS PROBLEMS
Which conic section is the
graph of the equation:
y2 =
y2/2
x2 +
y2/2
23x – 2
– x2/4 = 1
y2 = 1
+ x2/4 = 1
Questions?
New Topic
Counting stuff…
Sequences
1 2 3
What’s the next item in this
sequence?
Sequences
1 3
5
7
What’s the next item in this
sequence?
Sequences
Sequences are ordered lists
Sequences
They don’t have to be numbers
Sequences
→→
Sequences
Sequences
Each member of a sequence is
called a “term”
Sequences
Each term is designated by a
subscript:
a1, a2, a3, ..., an, ...
The 2nd term in the sequence
is a2
The nth term is an
Sequences
IN-CLASS PROBLEMS
What is a17 in the sequence:
1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22
23 24 25 26 27 28 29 30
Sequences
An infinite sequence goes on
forever:
a1, a2, a3, ...
A finite sequence stops at some
point:
a1, a2, a3, ..., a12
Sequences
IN-CLASS PROBLEMS
Which is the infinite series:
1 2 3 5 8 13 21 …
2 4 6 8 10
Sequences
Numerical sequences are often
formed by a formula that uses
one term to calculate the next
term in the sequence
Sequences
You evaluate a sequence by
plugging in the requested
numbers for each term
Sequences
IN-CLASS PROBLEMS
What is a3 if
n = 3 and an = 3 / n ?
Sequences
IN-CLASS PROBLEMS
What is a3 if
n = 3 and an = 3 / n ?
Plug in n=3!
Sequences
IN-CLASS PROBLEMS
What is a3 if
n = 3 and an = 3 / n ?
an = 3 / n
a3 = 3 / 3
a3 = 1
Sequences
IN-CLASS PROBLEMS
What is an if
n = 2
a1 = 4
an = an-1 + 1
Sequences
IN-CLASS PROBLEMS
What is an if
n = 2
a1 = 4
an = an-1 + 1
Plug in n = 2
Sequences
IN-CLASS PROBLEMS
What is an if
n = 2
a1 = 4
an = an-1 + 1
a2 = a2-1 + 1
a2 = a1 + 1
What is a1?
Sequences
IN-CLASS PROBLEMS
What is an if
n = 2
a1 = 4
an = an-1 + 1
a2 = a2-1 + 1
a2 = a1 + 1
a2 = 4 + 1
a2 = 5
Sequences
These are called “recursion
formulas” - the formula
calculates each term based on
the previous term
Sequences
(You have to be given a starting
term)
Questions?
Series
If you combine the numbers in
a sequence into a single
number, it’s called a “series”
Series
IN-CLASS PROBLEMS
What is the sum of the first
10 counting numbers?
Series
IN-CLASS PROBLEMS
What is the sum of the first
10 counting numbers?
1+2+3+4+5+6+7+8+9+10
Series
IN-CLASS PROBLEMS
Here’s a trick:
1 + 10 = 11
2 + 9 = 11
3 + 8 = 11
4 + 7 = 11
5 + 6 = 11
So the answer is 5 * 11 = 55
Series
It’s math class… so there’s
always a new symbol!
∑
means “add ‘em up!”
Series
We could have written:
What is the sum of the first
10 counting numbers?
as:
∑1 2 3…10
Series
We usually use ∑ with a
formula:
10
∑ n
n=1
This means “add up the n’s
where n goes from 1 to 10”
Series
IN-CLASS PROBLEMS
What is
4
∑ (n + 1)
n=1
Series
IN-CLASS PROBLEMS
What is
4
∑ (n + 1)
n=1
Just start adding up the (n + 1)s
Plugging in n=1, n=2, n=3, n=4
Series
IN-CLASS PROBLEMS
What is
4
∑ (n + 1)
n=1
= 1 + 1
+ 2 + 1
+ 3 + 1
+ 4 + 1
= 2 + 3 + 4 + 5 = ?
Series
Always new symbols!
Factorial notation
n! = n(n-1)(n-2)...3 * 2 * 1
Series
So, 4! would be:
4 * (4-1) * (4-2) * (4-3)
Don’t go to 4-4 (that would be
zero…)
4! = 4 * 3 * 2 * 1 = ?
Series
IN-CLASS PROBLEMS
What is 9!
?
What is 5!
?
Series
IN-CLASS PROBLEMS
What is 9!/5!
?
Questions?
Arithmetic Sequences
1 2 3
What’s the next item in this
sequence?
Arithmetic Sequences
1 3
5
7
What’s the next item in this
sequence?
Arithmetic Sequences
1 4
7
10
What’s the next item in this
sequence?
Arithmetic Sequences
These are called “arithmetic
sequences”
air-rith-MEH-tic
not
ah-RITH-meh-tic
Arithmetic Sequences
Each term in the sequence
(after the first) differs from
the preceding one by a constant
amount (positive or negative)
Arithmetic Sequences
start with a1
increase each time by "d"
Arithmetic Sequences
IN-CLASS PROBLEMS
start with a1
increase each time by "d"
3 7 11 15 19 …
What is a1?
What is d?
Arithmetic Sequences
start with a1
increase each time by "d"
General term of an arithmetic
sequence:
an = a1 + (n-1)d
Arithmetic Sequences
IN-CLASS PROBLEMS
an = a1 + (n-1)d
If a1 = 0 and d = 3 what are
the first four terms of the
arithmetic sequence?
Just plug in a1 and d
and n= 1,2,3,4
Arithmetic Sequences
IN-CLASS PROBLEMS
an
an
a1
a2
a3
a4
=
=
=
=
=
=
a1 + (n-1)d
0 + (n-1)3
0 + (1-1)3 =
0 + (2-1)3 =
0 + (3-1)3 =
0 + (4-1)3 =
?
?
?
?
Sequences
You can calculate the 10,000th
term in an arithmetic sequence
using the formula without having
to list up the 9,999 that come
before it!
Arithmetic Sequences
IN-CLASS PROBLEMS
If a1 = 5 and d = 2 what are
the first four terms of the
arithmetic sequence?
Arithmetic Sequences
IN-CLASS PROBLEMS
If a1 = 5 and d = 2 what are
the first four terms of the
arithmetic sequence?
an = a1 + (n-1)d
Arithmetic Sequences
IN-CLASS PROBLEMS
If a1 = 5 and d = 2 what are
the first four terms of the
arithmetic sequence?
an = 5 + (n-1)2
n = ?
Arithmetic Sequences
IN-CLASS PROBLEMS
If a1 = 5 and d = 2 what are
the first four terms of the
arithmetic sequence?
a1
a2
a3
a4
=
=
=
=
5
5
5
5
+
+
+
+
(1-1)2
(2-1)2
(3-1)2
(4-1)2
=
=
=
=
Arithmetic Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the arithmetic
sequence 1, 4, 7…?
Arithmetic Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the arithmetic
sequence 1, 4, 7…?
an = a1 + (n-1)d
What is a1?
What is d?
What is n?
Arithmetic Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the arithmetic
sequence 1, 4, 7…?
a1
a2
a3
a4
a5
=
=
=
=
=
1
1
1
1
1
+
+
+
+
+
(1-1)3
(2-1)3
(3-1)3
(4-1)3
(5-1)3
=
=
=
=
=
Arithmetic Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the arithmetic
sequence 1, 4, 7…?
a1
a2
a3
a4
a5
=
=
=
=
=
1
1
1
1
1
+
+
+
+
+
(1-1)3
(2-1)3
(3-1)3
(4-1)3
(5-1)3
=
=
=
=
=
1
4
7
10
13
Arithmetic Sequences
IN-CLASS PROBLEMS
So the sum is:
1+4+7+10+13 = 14*2 + 7 = 35
Arithmetic Sequences
Sum of the first n terms of an
arithmetic series:
Sn = n (a1 + an)
𝟐
For ours: S5 = (1 + 13) = 35
5
𝟐
Geometric Sequences
What is the next term in this
sequence:
1
2
4
8
Geometric Sequences
How about this one?
1
3
9
27
Geometric Sequences
Geometric sequences - each
term in the sequence (after the
first) is a common multiple
(positive or negative) of the
previous term
Geometric Sequences
For a geometric sequence, you
need:
the starting value a1
the multiple r
Geometric Sequences
IN-CLASS PROBLEMS
For the sequence: 1 4 16 64
What is a1?
What is r?
Geometric Sequences
General term of a geometric
series:
an = a1r n-1
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 1 and r = 2, what are
the first four terms of the
geometric sequence?
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 1 and r = 2, what are
the first four terms of the
geometric sequence?
an = a1r n-1
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 1 and r = 2, what are
the first four terms of the
geometric sequence?
an = 1(2 n-1)
What is n?
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 1 and r = 2, what are
the first four terms of the
geometric sequence?
a1
a2
a3
a4
=
=
=
=
1(2 1-1)
1(2 2-1)
1(2 3-1)
1(2 4-1)
=
=
=
=
?
?
?
?
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 1 and r = 2, what are
the first four terms of the
geometric sequence?
a1
a2
a3
a4
=
=
=
=
1(2 1-1)
1(2 2-1)
1(2 3-1)
1(2 4-1)
=
=
=
=
1(1)
1(2)
1(4)
1(8)
=
=
=
=
1
2
4
8
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 2 and r = 2, what are
the first four terms of the
geometric sequence?
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 2 and r = 2, what are
the first four terms of the
geometric sequence?
an = a1r n-1
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 2 and r = 2, what are
the first four terms of the
geometric sequence?
an = 2(2 n-1)
What is n?
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 2 and r = 2, what are
the first four terms of the
geometric sequence?
a1
a2
a3
a4
=
=
=
=
2(2 1-1)
2(2 2-1)
2(2 3-1)
2(2 4-1)
=
=
=
=
?
?
?
?
Geometric Sequences
IN-CLASS PROBLEMS
If a1 = 2 and r = 2, what are
the first four terms of the
geometric sequence?
a1
a2
a3
a4
=
=
=
=
2(2 1-1)
2(2 2-1)
2(2 3-1)
2(2 4-1)
=
=
=
=
2(1)
2(2)
2(4)
2(8)
=
=
=
=
2
4
8
16
Geometric Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the geometric
sequence 1, 4, 16…?
Geometric Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the geometric
sequence 1, 4, 16…?
an = a1r n-1
What is a1?
What is r?
Geometric Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the geometric
sequence 1, 4, 16…?
an = 1(4 n-1)
What is n?
Geometric Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the geometric
sequence 1, 4, 16…?
a1
a2
a3
a4
a5
=
=
=
=
=
1(4 1-1)
1(4 2-1)
1(4 3-1)
4-1
1(4 )
1(4 5-1)
=
=
=
=
=
?
?
?
?
?
Geometric Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the geometric
sequence 1, 4, 16…?
a1
a2
a3
a4
a5
=
=
=
=
=
1(4 1-1)
1(4 2-1)
1(4 3-1)
4-1
1(4 )
1(4 5-1)
=
=
=
=
=
1
4
16
64
256
Geometric Sequences
IN-CLASS PROBLEMS
What is the sum of the first
five terms in the geometric
sequence 1, 4, 16…?
1 + 4 + 16 + 64 + 256
= 341
Geometric Sequences
Sum of the first n terms of a
geometric series:
Sn =
For ours: S5 =
=
𝒂𝟏 (𝟏−𝒓𝒏 )
𝟏−𝒓
𝟏(𝟏−𝟒𝟓 )
𝟏−𝟏𝟎𝟐𝟒
=
𝟏−𝟒
𝟏−𝟒
−𝟏𝟎𝟐𝟑
= 341
−𝟑
Geometric Sequences
IN-CLASS PROBLEMS
Which is geometric?
3 6 9 12
1 2 4 8
7 10 13 16
2 3 4.5 6.75
2 4 6 8
Questions?
Fundamental Counting Principle
the number of ways things can
occur
Fundamental Counting Principle
Male/Female
and
Tall/Short
How many ways can these
characteristics combine?
Fundamental Counting Principle
Male/Female
and
Tall/Short
I try to build a tree:
Fundamental Counting Principle
Male/Female
Male
/
\
Tall
Short
and
Tall/Short
Female
/
\
Tall
Short
Fundamental Counting Principle
IN-CLASS PROBLEMS
Male
/
\
Tall
Short
Female
/
\
Tall
Short
4 possible ways to combine the
characteristics:
MT MS FT FS
Fundamental Counting Principle
How about:
Blonde/Brunette/Redhead and
Blue eyes/Green eyes/Brown eyes
Build a tree!
Fundamental Counting Principle
IN-CLASS PROBLEMS
Blonde
/ | \
Bl Br Gr
Brunette
/ | \
Bl Br Gr
Red
/ | \
Bl Br Gr
How many ways to combine these
characteristics?
Fundamental Counting Principle
IN-CLASS PROBLEMS
Blonde
/ | \
Bl Br Gr
Brunette
/ | \
Bl Br Gr
Red
/ | \
Bl Br Gr
How many ways to combine these
characteristics? 9:
BdBl BdBr BdGr BtBl BtBr BtGr
RdBl RdBr RdGr
Fundamental Counting Principle
The number of ways in which
characteristics can be combined
is found by multiplying the
possibilities of each
characteristic together
Fundamental Counting Principle
IN-CLASS PROBLEMS
Two pairs of jeans: black blue
Three shirts: white yellow blue
Two pairs of shoes: black brown
How many different ways can
you get dressed?
Fundamental Counting Principle
IN-CLASS PROBLEMS
Two pairs of jeans: black blue
Three shirts: white yellow blue
Two pairs of shoes: black brown
How many different ways can
you get dressed?
2 * 3 * 2 = 12
Fundamental Counting Principle
IN-CLASS PROBLEMS
Multiple choice quiz
10 questions
4 choices on each
How many ways are there to
answer the questions on the
test?
Fundamental Counting Principle
IN-CLASS PROBLEMS
Multiple choice quiz
10 questions
4 choices on each
4 * 4 * 4 *… (10 of them)
Fundamental Counting Principle
IN-CLASS PROBLEMS
Multiple choice quiz
10 questions
4 choices on each
4 * 4 * 4 *… (10 of them)
Otherwise known as 410
= 1,048,576
Fundamental Counting Principle
IN-CLASS PROBLEMS
Multiple choice quiz
10 questions
4 choices on each
How many ways out of the
1,048,576 can you get a 100?
Fundamental Counting Principle
IN-CLASS PROBLEMS
Multiple choice quiz
10 questions
4 choices on each
1/1,048,576 chance of getting
100% if you guess on all
questions
Fundamental Counting Principle
IN-CLASS PROBLEMS
How many zip codes?
5 slots
Can’t start with a 0 or a 1
Fundamental Counting Principle
IN-CLASS PROBLEMS
How many zip codes?
___ ___ ___ ___ ___
Fundamental Counting Principle
IN-CLASS PROBLEMS
How many zip codes?
8
10
10
10
10
Fundamental Counting Principle
IN-CLASS PROBLEMS
How many area codes?
8
10
10
8*104 = 80,000
10
10
Fundamental Counting Principle
IN-CLASS PROBLEMS
In Canada, they alternate
Letter Number Letter
Number Letter Number
How many area codes can they
have?
Fundamental Counting Principle
IN-CLASS PROBLEMS
In Canada, they alternate
Letter Number Letter
Number Letter Number
26*10*26 * 10*26*10
= 263 * 103 = 17,576,000
(A whole lot more than 80,000!)
Questions?
Liberation!
Be sure to turn in your
assignments from last week
to me before you leave
Don’t forget
your homework
due next week!
Have a great
rest of the week!