Almost forgot to do a blog again! Better late than never,right? Okay, this week I'll be reviewing permutations and combinations. Permutations and combinations are formulas used to find different ways something can be ordered. The difference between a permutation and combination is that you use a permutation when the order is important and you use a combination when the order is NOT important. Some things to know: The formula for a permutation is n!/(n-r)! The formula for a combination is n!/(n-r)r! Your n will ALWAYS be larger than you r. Let's do some examples to help you better understand. Example 1: 5P3 5P3=5!/(5-3)! =5!/2! =5*4*3*2*1/2*1 = 60 Example 2: 5C3 5C3=5!/(5-3)3! =5!/2!3! =5*4*3*2*1/2*1*3*2*1 =20/2 =10 And that's all there is to it.

almost forgot to do another blog!!

well im BUKU tired and i cant believe i just took out my laptop to do this.
i saw somebody write the trig chart i think so thats what im doing. i can barely type im so tired

sin 0= 0

sin pi/6=1/2

sin pi/4= square root of 2/2

sin pi/3= square root of 3/2

sin pi/2= 1 



cos 0= 1

cos pi/6= square root of 3/2

cos pi/4= square root of 2/2

cos pi/3= 1/2

cos pi/2= 0




csc 0= undefined

csc pi/6= 2

csc pi/4= square root of 2

csc pi/3= 2 square root of 3/3

csc pi/2= 1




sec 0= 1

sec pi/6= 2 square root of 3/3

sec pi/4= square root of 2

sec pi/3= 2

sec pi/2= undefined




tan 0= 0

tan pi/6= square root of 3/3

tan pi/4= 1

tan pi/3= square root of 3

tan pi/2= undefined




cot 0= undefined

cot pi/6= square root of 3

cot pi/4= 1

cot pi/3= square root of 3/3

cot pi/2= 0

who doesnt love the trig chart? idk but i better get credit because even tho it looks like i did nothing, this was the hardest blog yet considering how tired i am!

goodnight bloggers

you'd think after all this time i'd remember to blog before 10:00 sunday night...you'd be wrong.

This week I am going to review the oh so exciting concept that is multiplying matrices. Pay attention…this is a toughy.

In order to multiply matrices, the dimensions have to be precise. You must have the same number of columns in the first matrix as rows in the second matrix. If you do not have such dimensions, YOU CANNOT MULTIPLY. If you do not have proper dimensions, the answer is not defined. DO NOT CONFUSE THIS WITH UNDEFINED. Undefined means you have divided by zero. NOT DEFINED means no solution.

-Helpful hint, after multiplying, your answer matrix will have the same number of rows as the first matrix and columns of the second matrix.

Example:
[3 4] [2 3 1] [22 33 23]
[6 7] X [4 6 5] = [40 60 41]
[9 1] [22 33 14]

Since the dimensions of the first matrix are 3x2 and the dimensions of the second matrix are 2x3, your answer matrix will be a 3x3.

First, multiply row one by column one. (I’m going to fill answers into the matrix as I go)
3(2)+4(4)=22

First row second column, then first row third column.
3(3)+4(6)=33 3(1)+4(5)=23

Continue the same process with rows two and three.

That is all.

--Sarah

Review of De Moivre's Theorem!

Soo today, we are going to learn all about De Moivre’s theorem! As I said in an earlier blog, chapter 11 consists of a few formulas to be followed. Most of them are really easy. (Especially in this section because there is only one that you must know J) It is pretty simple if you follow the theorem exactly how it is stated. There is one thing that you need to keep in mind throughout this section and that is:

**DO NOT DRAW ARGAND DIAGRAMS**


De Moivre's theorem states the 2=rcisø then z^n=r^n cis(nø)


So now for the emphasis example!


Evaluate (2sin45)^2

z^2=2^2cis2(45)
z^2=4cis90

Since I don't think I hit a 150 words yet, I'll do a problem like this but working backwards.


z=4cis20º (Use De Moivre's theorem to find z^3)

z^3=(4)^3cis(3(20))
z^3=64cis60

So basically if you know and learn De Moivre's theorem, you can work any of these problems.


Hope you learned something!

Carleyyyy :) have a greatttt week!!

11-2

Things you should know:

• z = x + yi
• z = rcostheta + rsintheta i 
• z = rcistheta
• |z |= square root of x^2 + y^2
• To multiply complex numbers:

1. Foil for rectangular problems.
2. Multiply r and add theta for polar problems.

Example: 0 + i

What form is it in? Rectangular form.

Find r and theta.

r = square root of 0^2 + 0^2. 

r = 0 + 0 =0

theta = tan^-1(0)

 Theta = 0

When you put that on the quadrants, you will get 180 and 360.

So once you do that, your answers will be square root of 2 cis 180 degrees and -square root of 2 cis 360 degrees.

Example 2: 10 cis 20 degrees

• What form is it in? Polar form.
• Solve for x and y.
• x = 10cos20= 9.367
• y=10sin20= 3.420
• Final answer: 9.367 + 3.420

-Amber :P

13-1 Review

This week I am going to review with you all the information that we learned in chapter 13, section 1. In chapter 13, section 1, we learned about arithmetic and geometric sequences. In this section, we just learned the basics of geometric and arithmetic sequences. Here are the notes.

Notes:
• An arithmetic sequence is a sequence that is generated by adding the same number each time.
• The formula for an arithmetic sequence is as follows: tn = t1 + (n – 1) d
• A geometric sequence is a sequence that is generated by multiplying the same number each time.
• The formula for a geometric sequence is as follows: tn = t1 * r^(n – 1)
• n = number of terms
• tn = term number
• t1 = term 1
• d = what is added
• r = what is multiplied

Example: Identify the following as an arithmetic or geometric, and find the formula for the nth term: 3, 6, 9, 12, ……
• this is an arithmetic sequence
• tn = 3 + (n-1) 3
• tn = 3 + 3n-3
• tn = 3n




-Braxton-

13-4

today im going to teach you how to do infinite limits. this is a review because i taught this before and we went over it. some key notes to know is that if:
1) (degree)top=(degree)bottom then limit = lead coeff / leading coeff.
2) (degree)top > (degree)bottom then limit = +/- (infinity)
3) (degree)top < (degree)bottom then limit = 0
* if it doesn't follow rules then yoy plug into y= in calculator, 2nd table, plug in 10|100|1000|10000
until you see a pattern*
4) E +ve # = + or - inf
    E -ve # = 0
*if it is geometric & |R| < 1 then limit = 0. if |R| > 1 the limit = (infinity)

EX's:
1) lim/ n (infinity)   n^3 + 2n^2 + 6 / n^2 - 4n^3
so (degree)top = (degree)bottom = -1/4

2) lim/ n (infinity)  n^3 + 2n^2 + 6 / n^2
 so (degree)top > (degree)bottom = +(infinity)

3) lim/ n (infinity)  5n - 5 / n^2
so (degree)top < (degree)bottom = 0

Review of the Trig Chart

So since all we did this week was take tests and review for some more tests. I am going to just review the oh so wonderful and important trig chart.

sin 0= 0

sin pi/6=1/2

sin pi/4= square root of 2/2

sin pi/3= square root of 3/2

sin pi/2= 1 



cos 0= 1

cos pi/6= square root of 3/2

cos pi/4= square root of 2/2

cos pi/3= 1/2

cos pi/2= 0




csc 0= undefined

csc pi/6= 2

csc pi/4= square root of 2

csc pi/3= 2 square root of 3/3

csc pi/2= 1




sec 0= 1

sec pi/6= 2 square root of 3/3

sec pi/4= square root of 2

sec pi/3= 2

sec pi/2= undefined




tan 0= 0

tan pi/6= square root of 3/3

tan pi/4= 1

tan pi/3= square root of 3

tan pi/2= undefined




cot 0= undefined

cot pi/6= square root of 3

cot pi/4= 1

cot pi/3= square root of 3/3

cot pi/2= 0

you also might need to know these few things.

pi/6= 30 degrees, pi/4= 45 degrees, pi/3= 60 degrees, pi/2= 90 degrees, sin=sine, cos=cosine, csc= cosecant, sec=secant, tan=tangent, cot=cotangent.

so that's it. later
Brad

Review of 11-2

So this week I am going to do my blog on a review of section 11-2. This section is on problems with complex numbers. This should be very simple and easy to remember. So I'll get started by giving you a few formulas.

• z=x+yi
• z=rcos theta + rsin theta i
• z=rcis theta
• |z|=square root of x^2 + y^2 

Note: To multiply complex numbers:

• With rectangular problems remember to always foil.
• With polar problems you will multiply r and add theta.

okay, so now I'll do an example for you. I might even do two examples, but it depends.

Example: -1 + i

Since it is in rectangular form, you are going to find r and theta.

r=square root of -1^2 + 1^2. Which equals square root of 2.

theta=tan^-1(-1). Which equals 45.

When you put that on the quadrants, you will get 135 and 315.

So once you do that, your answers will be square root of 2 cis 135 degrees and -square root of 2 cis 315 degrees.

Example 2: 6 cis 100 degrees

• x=6 cos 100= -1.042
• y=6 sin 100= 5.909
• Your answer is going to be -1.042 + 5.909

so, that's it for this week. Hope I helped you remember how to work these problems. byee.

--Halie!

Ch. 13 Review

13-1
This week I'm going to do a review on 13-1 where we learned about arithmetic and geometric sequences. We learned how to identify a sequence as arithmetic or geometric, how to find the nth term in a sequence, and how many terms are in a sequence. We also learned how to find the mean for both types of sequences.
-arithmetic sequence: sequence that is generated by adding the same number each time
  formula: tn = t1 + (n-1)d
-geometric sequence: sequence that is generated by multiplying the same number each time 
  formula: tn = t1 x r^(n-1)
-arithmetic mean: a + b/2
-geometric mean: square root of ab


Example 1: Identify the following as arithmetic or geometric and find the formula for the nth term.
4,8,16,32,...
-geometric
-tn = t1 x r^(n-1)
        4 x 2^(n-1)
        4 x 2^n x 2^-1
        4 x 2^n/2
        = 2 x 2^n


Example 2: Find the indicated term of the arithmetic sequence.
t1 = 3, t4 = 12, t30 = ?
- 3 + d + d + d = 12
  3 + 3d = 12
  3d = 9
   d = 3
-t30 = 3 + (30 - 1) (3)
 = 90

Trig Reviewwww

Today we are going to do some review from the trigonometry section. If you do remember Trig, you are good to go and can probably work my examples right now. But incase you don’t remember everything; all you need to know right now in trig is basically formulas as well as the trig chart. For starters, you must remember:

· Sin=1/csc

· Cos=1/sec

· Csc=1/sin

· Sec=1/cos

· Tan=1/cot

· Cot=1/tan

· Tan=sin/cos

Apart from knowing that, you must also know the Pythagorean Identities. To refresh your memories, these are the Identities you will need to use when doing a lot of trigonometry.

· Sin^2+cos^2=1

· 1+tan^2=sec^2

· 1+cot^2=csc^2

Now that we’ve went over some of the basic things needed to work the examples I am about to show you, we can begin J

Example 1: Simplify:

(Sinx^2+Cosx^2)/ Sinx

1. Look for identities. You have one, so simplify it.

2. You now get 1/Sinx. You can simplify this by referring to the notes above.

3. You get Cscx as your final answer J

I hope this review helped!

CARLEYYYYY

degrees to radians

Since part two of our giant trig exam is tomorrow I figured I’d keep it simple and review how to convert degrees to radians and vice versa.

You are going to multiply 60 by pi/180.

All you really have to do is simplify 60/180 then add pi in.

You’re answer should therefore be 1pi/3, which is simply pi/3

Now for radians to degrees.

Convert 5pi/4 to degrees.

To convert radians to degrees the process is almost exactly the same

You multiply 5pi/4 times 180/pi.

The pi cancels leaving you with 5/4 times 180.

5 X 180= 900/4=225 degrees.

If you do not get a whole number, you need to convert to degrees minutes and seconds.

To convert to degrees minutes and seconds you multiply the number behind the decimal by 60. This number becomes minutes. If there is another set of numbers behind the decimal, multiply by 60 again. If you still don’t have a whole number after multiplying by sixty twice, you drop the number behind the decimal and the number in front of the decimal becomes seconds.

YOU WILL GET POINTS OFF IF YOU DO NOT CONVERT.

Good luck to everyone on the tests this week

--Sarah

7-1

One can measure angles in either degrees or radians. It really depends on whether the problem states it in degrees>0>degrees or rads>0>rads

1) Breaking a problem into degrees, minutes, and seconds

If one is converting degrees to minutes, all of the numbers that are behind the decimal have to multiplied by 60
-If one is converting degrees to seconds, all of the numbers that are behind the decimal have to multiplied by 60. 
-If one is to convert minutes and seconds back to degrees, then use the following equation: 

degrees+ (min/60)+ (sec/3600)

Ex 1: Convert 76.43 degrees to degrees, minutes, and seconds.

a) .43 X 60 = 25.8'
b) .8 X 60 = 48"
answer = 76 degrees 25' 48"

2) Converting from degrees to radians and radians to degrees

-To convert degrees to radians: degree(times)(pi/180)

-To convert from radians to degrees: radianspi (times)180/pi)

*pi will cancel out

Ex 2: Convert 235 degrees to radians.

a) 235 X (pi/180)

answer = (47/36)pi

-Sameer

10-2

Today I'm go to explain how to do chapter 10, section 2. This is the sum and difference for tangent. There's only two formulas for this secction. They're almost the same thing except the sign changes. Formulas: tan(alpha+beta) = tan(alpha) + tan(beta)/1-tan(alpha)tan(beta) tan(alpha-beta) = tan(alpha) - tan(beta)/1+tan(alpha)tan(beta) REMINDER: You do not plug in for formulas like the ones above, you replace. Okay, time for some examples!!!! Suppose tan alpha = 1/3 and tan beta = 1/2 Find tan(alpha+beta) = (1/3 + 1/2)/(1-(1/3)(1/2)) =( 2/6+3/6)/(1-1/6) = (5/6)/(5/6) = 1 Suppose tan alpha = 4/3 and tan beta = -1/2 Find tan(alpha+beta) = (4/3+(-1/2))/(1-4/3(-1/2) =(8/6+(-3/6))/(1-(-4/6) =(5/6)/(10/6) =1/2

8-4 Review

So this week we have the lovely trig exam, which I am just sooo excited for. SO since all we did was review and review AND review, I am going to just do my blog on 8-4.

8-4 is on the relationship among functions. This should be pretty easy. But before I can start with examples, i am going to give you a few notes that you need to know.

• The first thing you do is to do all the possible algebra to the problem.
• Once you do everything possible, you would first try to use your pythagorean identities.
• After that you would move everything to sin and cos.
• The next thing you would do is do algebra again.
• Once you do all of that you keep repeating steps 1 through 3 until your problem is completely simplified.

Now you need some formulas.

• cscx=1/sinx
• tanx=sinx/cosx
• cotx=cosx/sinx
• secx=1/cosx
• sin^2x+cos^2x=1
• 1+tan^2x=sec^2x
• 1+cot^2x=csc^2x

Well I guess now that you have all of what you need I'll do an example or two.

Example:  cos^2x+sin^2x

• No algebra can be done.
• So then you look for identies you can use, which this problem is one which means it will equal to 1.
• So you answer is going to be 1.

So that's it for this week. Later

Brad

Chapter 9 Review

Well, this week is the trig test so I thought it would be a good idea to review a trig chapter. This week I am going to review with you all the information that we learned in Chapter 9. Chapter 9 is all about triangles. There are a few formulas that you need to know for this chapter.

Notes:
• SOHCATOA: sin (theta) = opp. / hyp. cos (theta) = adj. / hyp. tan (theta) = opp./adj.
• To find the area of a right triangle, use the formula A=½ bh
• To find the area of a non right triangle, use the formula A= ½ (adj.) (adj.) sin(angle b/w)
• Law of Sines: (sin A / a) = (sin B / b) = (sin C / c)
• Law of Cosines: opp. leg^2 = (adj. leg^2) + (other adj. leg^2) – 2(adj. leg) (other adj. leg) cos (angle b/w)

Example: Find the area of non right triangle ABC when: AB=4, BC=6, and B=60 degrees
• ½ (4) (6) sin (60 degrees)
• 12 sin (60 degrees)
• A=10.392 u^2


-Braxton-

11-2

today im reteaching 11-2 which is complex numbers with polar and rectangular. the complex form of rectangular is z= x + yi. the complex form for polar is z= r cis (theta). you can also multiply these numbers and to do so for rectangular you would just do FOIL and get your answer. for polar you mulitply the r's and add the theta's. here is an example below.

EX:
express each complex number in polar form
1) -1 + i
you do same steps to convert from rectangular to polar.
(the square root of) (-1) ^2 + (1)^2 = (the square root of) 2
(theta) = tan (inverse) 1/-1
the quadrants that they are negative is 2nd and 4th
(theta) = tan (inverse) 45
convert to second and fourth quadrant
-45 +180= 135
-45 +360= 315
now you have to figure out which to use
(-1,1) is in second quadrant so you use 135
so your polar form is r cis (theta) = (the square root of) 2 cis 135 (degrees)

Review of Chapter 7

Okayy, so this week we reviewed for the trig test! So I am going to do my blog on a review of chapter 7. This is should be very easy because you should already know all of this. So lets get started! First, I am going to give you some formulas.

• K=1/2r^2 Ɵ
• K=1/2rs
• s=rƟ

In the last formula, r=distance between two objects, Ɵ=apparent size, and S=diameter of an object.

Okay, so now I am going to give you a few examples.

Example 1: A sector of a circle has a radius 8 cm and central angle 2 radians. Find its arc length and area.

In this problem,

• R(radius)=8cm
• Ɵ(central angle)=2
• K(area)=?
• S(arc length)=?

To solve this problem, you would use the equation k=1/2r^2Ɵ.

Therefore, k=1/2(8)^2(2) so k=64cm^2.

 

You then plug into k=1/2rs. Since you’re solving for s, it becomes k/1/2r=s.
Therefore s=64/4 so s=16cm.And those are your two answers!

 

Well, that's it for this weeeek. See ya later! Byeee.

 

--Halie!

11-1

11-1 Polar
This week I'm going to do a review on 11-1 where we learned how to convert from polar to rectangular and from rectangular to polar. In this section, we also use polar points in which we don't use (x,y) but (r,theta) instead.
-To convert from polar to rectangular: x = r(cos)(theta)    y = r(sin)(theta)
-To convert from rectangular to polar: r = square root of x^2 + y^2     theta = tan^-1(y/x)

Example 1: Give the polar coordinates for (5,0).
r = square root of 5^2 + 10^2
r = +/-5
theta = tan^-1(0/5)
theta = 0
final answer: (5,0) (-5,0)


Example 2: Give the rectangular coordinates for (3,30 degrees).
x = r(cos)(theta)
x = 3cos30
x = 3(sq. root of 3/2)
x = 3(sq. root of 3)/2
y = r(sin)(theta)
y = 3sin30
y = 3(1/2)
y = 3/2
final answer: (3(sq. root of 3)/2, 3/2)

12-5

Similar to what was learned in lesson 12-2 through 12-4, the only thing that changes is the number of variables present within the vector. It is pretty simple to do and not at all complicated with the addition of a variable.

Now there's is

- a x, y, and z

Ex 1: Simplify <2, 5, -4> + 3<2, 4, -1>

Distribute the THREE to the second vector

<2, 5, -4> + <6, 12, -3>

Now just add the terms corresponding in the first vector to the second vector (x1+x2, y1+y2, etc.)

The answer is <8, 17, -7>

-Sameer