Late reflection #28

here's an example:
I'll review Sequences and Series:

let's start with formula review:

arithmetic sequence:

tn=t1+(n-1)d

*d being your ratio, or number that's added or subtracted.
*t1 being your first term
*n being the term in the sequence we're trying to find


ex:
In the arithmetic sequence:3,5,7,9-- find the 28th term.

t28=3+(27)(2)
t28=3+54
t28=57

geometric sequence, or a sequence with multiplication and division:

tn=t1*r^(n-1)

*t1 is first term.
*r is your ratio, multiplied or divided by.
*missing term.

ex:
In the geometric sequence: 2,4,8,16-- find the 10th term

t10= 2*2^9
t10= 2*512
t10= 1024


1.) Find the sum of the first 16 terms of the arithmetic series 7+11+15+19+...
*Since it says it's an arithmetic series, you will use the arithmetic sums formula>> sn = n(t1 + tn)/2
*you just plug the numbers into the formula. "n" is 16 because that is the indicated number of terms.
*t1 is the first term in the sequence, which is 7.
*Now to find "tn" you have to plug your numbers into the first arithmetic formula we learned:
tn=t1+(n-1)d
*So you get tn=7+(16-1)(4)
(4 is the difference because that is what's being added each time)
*So tn=67
*Now you can plug that into your sum formula
*So you get s16=16(7+67)/2
*So your final answer is 592

Late reflection #27

Here's my second one.

In the world of Flatland, classes are distinguished using the "Art of Hearing", the "Art of Feeling" and the "Art of Sight Recognition". Classes can be distinguished by the sound of one's voice, but the lower classes have more developed vocal organs, enabling them to have the voice of a polygon or even a circle. Feeling, done by the lower classes and women, determines the configuration of a person by feeling one of their angles. The "Art of Sight Recognition", done by the upper classes, is aided by "Fog", which allows an observer to determine the depth of an object. With this, polygons with sharp angles relative to the observer will fade out more rapidly than polygons with more gradual angles. The population of Flatland can "evolve" through the Law of Nature, which states: "a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale of development and nobility. Thus the son of a Square is a Pentagon; the son of a Pentagon, a Hexagon; and so on."

It's just as in our world, our parents want us to grow up to have more than they had before. To keep the family name building, never flatlining or eventually burning out.

Late reflection #26

For some reason I just realized that i've missed a million blogs!
I'll start from here and go on...

I'll do a blog about the book Flatland:

In the book, men are portrayed as polygons whose social class is directly proportional to the number of sides they have; therefore, triangles, having only three sides, are at the bottom of the social ladder and are considered generally unintelligent, while the Priests are composed of multi-sided polygons whose shapes approximate a circle, which is considered to be the "perfect" shape. On the other hand, the female population is comprised only of lines, who are required by law to sway back and forth and sound a "peace-cry" as they walk, because when a line is coming towards an observer in a 2-D world, it appears merely as a point. The Square talks of accounts where men have been killed by being stabbed by women. This explains the need for separate doors for women and men in buildings. Also, colors in Flatland were banned, when lower classes painted themselves to appear to be higher ordered.

Reflection

I'm so glad i dont have to go to school friday, so i'll do this blog early. Here is some review stuff from chapter 8. We learned how to find the angle of inclination, which i found was really easy compared to some stuff we learn. We also learned about amplitudes, periods, vertical shifts, etc. There are some formulas we had to learn to be able to work these problems:

1.) For a line
m=tan alpha where m=slope and alpha=angle of inclination
2.) For a conic
tan 2 alpha=B/A-C
3.) For a conic if A=C then
a=pi/4

EXAMPLE:

Find the angle of inclination.
2x+5y=15
m=-2/5 tan alpha=-2/5 Checks are in the II and IV area and 21.801 degrees in I
alpha=tan^-1(-2/5)
180-21.801 alpha ~ 158.199 degrees, 338.199 degrees
158.199+180

Comments

Here are two examples on sequences

Find the formula for the nth term of the arithmetic sequence. 3,5,7...
tn=3+(n-1)(2)
tn=3+2n-2
tn=1+2n

Find the formula for the nth term of the sequence. 3,4.5,6.75...
r=4.5/3=3/2
6.75/4.5=3/2

tn=3x(3/2)^n-1

late blog #5

here's some review from when we learned how to work trigonometry, use the trigonometry chart, the unit circle, and trigonometry inverses. The only thing that I understood this week was how to use the unit circle.Here is an Example: (-3, 4) find all six trigonometry functions-draw your graph and draw a line from the origin to the point that is given.-then use the chart we were given in class and label all six of the trigonometry functions.-your answer would be:Sin =4/5Cos =-3/5Tan =-4/5Csc =5/4Sec =-5/3Cot =-3/4The one thing I didn’t understand is how to use the trigonometry chart to solve problems. For example: find the reference angle and the exact answer. How do you determine what to use to solve the problem?

late blog #4

here's some review from when we learned how to simplify trigonometry functions, graph trigonometry functions, ideates and equations. The thing I did understood the most was how to graph a trigonometry function. Example:y=2sin (3x+pie)-4Amp. =2Period=2 pie/3 leave pie out p/4=1/6 add to each to each point.0Pie/6Pie/3Pie/22 pie/3phase shift=pie0-pie=-piePie/6-pie=-5pie/6Pie/3-pie=-2 pie/3Pie/2-pie=-pie/22 pie/3-pie=pie/3Y=2 sin (3x + pie)-4Then you draw your graphvertical shift=4The one thing I didn’t understand this week is how to solve a trigonometry function using the identities and equations. I don’t understand how you use the identities and the equations.

late blog #3

here's some review from when we learned about different kinds triangles, how to solve them, and how to find the area. The thing that I understood the most was using SOHCATOA to solve right triangles.Example:If you are given a triangle ABC, side a is 2, side b is 3 and you have to find angle B.You use tan theta=3/2Theta = tan^-1(3/2)Theta = 56.310 degreesThe thing that I didn’t quite understand was using the law of cos. How do you use the formula to solve?

late blog #2

here's some review this is when we learned how to solve circles, eclipses, Hyperbolas, and parabolas. The thing that I understood the most this week was how to solve for circles.Example:Find the intersection of the circleX^2+y^2=25 & y=2x-21. y=2x-22. x^2+(2x-2)^2=253. x^2+4x^2-8x+4=255x^2-8x+4=255x^2-8x-21=05x^2-15x+7x-21(x-3) (5x+7)X=3 x=-7/54. 2(3)-2=4 (3, 4)2(-7/5)-2=-24/5 (-7/5, -24/5)The thing that I didn’t understand was how to solve for hyperbolas. I don’t understand how you find the foci.

late blog #1

here's some reveiw. how to solve negative exponents, use logs, and log properties. This thing that I understood the most was how to solve logs.Example:Log 2 16=x2x=16 2x=2^4X=4Log 2 16=410^x=1020Log 1020=xLog 1020=3.009The thing that I don’t under stand from this week was how to solve negative exponents for example (a^-2 + b^2) ^-1/ (1/a^2 + 1/b^2) ^-1. How do you solve this I don’t understand?

Reflection 2/29/10

Here are some examples of easy logs. This chapter was easy but maybe some of yall have forgotten how to do these so I did a few examples.

Examples of some logs:

log 7 of 49 = 2 log x of 36 = 2
7^2 =49 x^2 = 36
x = 6

log 3 of 27 = 3 log x of 64 = 3
3^3 =27 x^3 = 64
x = 4

log 2 of 32 = 5 log x of 100 = 2
2^5 =32 x^2 = 100
x = 10

Now I get the main concept of solving for exponents but I am never sure when the problem is simplified all the way. There are so many little steps.

Reflection

Flatland.

The square ends up visiting lineland which is a one-dimensional world and everyone is either a line or dot. The dots are woman, Short lines are kids, and long lines are men. And in the middle of the line is the king. He then visits spaceland which is new to him. eventually he explains it to his grandchild and when he tries to explain it to everyone else he is sadly thrown in jail. Terrible ending.

Reflection

Back in the gap law of cosines an sines.....lets review!!

Law of Sines: sin(opp. angle)/Leg=sin(Opp. angle)/Leg. Set up a proportion.

Example. triangle ABC where A=36 degrees a=3 and B=56 degrees. find b

sin36/3=sin56/x 3sin56/sin36= x Plug into your calculator and done.

Law of Cosines:

(opposite leg)^2=(adjacent leg)^2+(other opposite leg)^2-2(leg)(leg)cos(angle in between)

example:
for a triangle with C=36 degrees a=5 b=6

c^2=5^2+6^2-2(5)(6)cos36
c=Square root of(25+36-2(5)(6)cos36)
c= 3.53

Are of non-right triangle:
1/2(leg)(leg)sin(angle between)

Example: Triangle ABC has sides a=5 b=3 and C=40 degrees

therefore your formula will be 1/2)(5)(3)(sin(40)) type that in your calculator and you will get the answer.

relfection

Inequalities. There are two types of inequalities, regular inequalities and absolute value inequaties. In absolute value inequalities there are two different types determined by the signs if it is not an equal sign. They can either be and/or inequalities. And inequalities always have a less than symbol and or inequalities always have a greater than symbol.

*In absolute value inequalities you always get two answers.

*Also remember if dividing by a negative the greater/less than sign will switch to the opposite.

Examples:

1. 4x + 1 > 13

*For this problem you only get only answer because it isn't absolute value.

*Subtract 1 from 13 and you get 4x > 12

*Divide by 4

*Answer x > 3

2. 3x - 4 + 5 <22

*Subtract the 5 over from the 27 and you get 3x - 4<22

*Solve the equation first by adding 4 over to all sides -18<3x<26

*Divide by 3 on all sides your final answer is -6<26/3

Reflection (got monday off!)

Well i'll blog about the book bein we finished it....personally, i didn't like it very much...at all. BUT, it did have some very good metaphors an deeper meanin about our social class in real life. I believe that the whole thing about how many sides you have showin your social class can be related to material possessions people have showin their social class. When you see a person in a suit drivin a 70 grand Beamer, with 300 Oakleys on you know that they guy either comes from money are has found his way into some money, so he's "socially highly ranked" because he's rich, whereas the guy in the 1998 F-150 with a tee shirt an 10 dollar walmart sunglasses on automatically get considered as "middle class or lower middle class." Another big metaphor i think is the way a shapes children gain one side. This means you are basically the same social class as your parents. In today's world, while many young men an women who come from a fairly poor background still sometimes make it to a high salary, for the most part you are born into a social class and stay within it your whole life.

To dig a little deeper i think the square goin to Lineland an gettin aggrivated with the king for not understandin Flatland, then gettin aggrivated with the Spaceland creature because he doesn't understand Spaceland is a metaphor for somewhat of hypocracy. When you understand somin an someone else doesn't, we get annoyed, but when we don't understand somin an someone else does we get aggrivated with them because we don't understand, which is in my mind bein a hypocrite.

**Jus a note, i thought the end was stupid, i was hopin for the end to be good, but i thought it was terrible personally.

reflection

sin 0= opp. leg/ hypotenuse

cos 0= adj. leg/ hypotenuse

tan 0= opp. leg/ adj. leg

when given a right triangle with two sides equaling 553 meter and 100 meters.. and you are trying to find angle A it is adjacent to the right angle... refer back to SOH CAH TOA. visualize it: angle a is opposite a side and adjacent a side so you would use tangent.

tanA= 553/100
A=tan inverse (553/100) Remember you are trying to find an angle
A= 79.750 degrees always use the inverse of tan, sin, or cos.

now when given a triangle with a side and an angle you need to find both sides x and y.

right triangle. hypotenuse = 20ft angle A= 70 degrees

look at the side opp. the hypotenuse.


side y. we know that opposite is 20 ft and adj is x so plug it in to the formula

cos 70=x/20
x=20cos70
approx. = 6.840/ft
side x. we know that opp. side is y and hypotenuse is 20 ft. the angle is 70 degrees so lets plug it in...
sin 70= y/20
y= 20sin70 approx. = 18.79 ft

reflection 28

Here is some stuff about circles because i just found my old notebook and i need to remember this stuf too:)

Equation of a circle in standard form is:
(x-h)^2+(y-k)^2=r^2
center(h,k) radius(r)
If not in standard form, you must complete the square to put into standard form.
If given the center and a point, you can use the distance formula to find the radius.
To find the intersection of a line and a circle:
first, solve for y.
second, substitute it in your circle equation.
third, solve for x.
fourth, plug y in to get y.
NOTE: if your x-value is imaginary there is NO INTERSECTION.
Ex:Find center and radius:
1. (x-3)^2+(y+7)^2=19
center: (3, -7)
radius: (square root of 19)
2. PUT IN STANDARD FORM TO FIND CENTER AND RADIUS:
x^2+y^2+16x-12y+5=0
COMPLETE THE SQUARE:
x^2+16x+y^2-12y=-5
x^2+16x+64+y^2-12y+36=-5+64+36
(x+8)^2+(y-6)^2=95
CENTER:(-8,6)
radius: (square root of 95)

Reflection 28.

So were still on the book flatland and it feels like we've been reading it forever. I find it a little confusing so hopefully the movie will clear everything up. Our trig exam is about to come up so i'll review something that's pretty simple from trig (Tanget formulas).

Sum Formula for Tangent
tan(alpha + beta) = tan alpha + tan beta/1-tan alpha (tan beta)

Difference Formula for Tangent:
tan(alpha - beta) = tan alpha - tan beta/1+tan alpha (tan beta)

EXAMPLES:

1)Find the exact value of tan15 degrees+tan30 degrees/1-tan15 degrees(tan30 degrees)
= tan(15 degrees + 30 degrees)
= tan(45 degrees)
= 1

*For this problem you know alpha and beta so you would just plug it into the formula. Once you've done that you figure out what is tan 45 on the trig chart, then that's your answer.

2)Find tan(alpha+beta)
tan alpha=2 tan beta=1
tan(alpha+beta) = tan alpha+tan beta/1-tan alpha(tan beta)
= 2+1/1-(2)(1)
= 3/-1
= -3

*For this remember to plug the number into the whole formula replacing tan alpha or tan beta. Not just tan, beta, alpha.

Reflection #28

Okay, I'm going to review domain and range:

You must remember the rules...
the only time you don't find the range is when the problem is a fraction.

When working with a polynomial, the domain will always be (-infinity,infinity) but for range, it depends: if the leading exponent is an odd number, the range is the same as the domain, but if the leading exponent is 2, then it's [vertex,infinity) if it's positive and if it's negative, it's (-infinity,vertex].
**The vertex is -b/2a

For square roots:
1)set the inside = 0
2)set up a # line
3)try values on either side (like graphing in Ch.2)
4)eliminate anything -ve
5)set up intervals (infinity, )( , )...
( ,infinity)
and for range, just graph

Now, for fractions:
1)set bottom = 0
2)solve for x
3)set up intervals

For absolute value:
the domain is always (-infinity,infinity) and the range is [shift,infinity) if positive and if negative its (-infinity,shift). (Shift is when the entire graph moves to however many spaces the problem implies such as {x+1}-1, the last -1 indicates that the graph moves on the y-axis to -1, so the range would be (-1,infinity).)

Would anyone like to add some examples?

reflection for 2/28

Sooo, we are finally done flatland and its a pretty interesting book, i can't wait to see the movie on tuesday. The questions on it this week weren't that hard, most of them were opinionated questions but i like those, haa. I think after we are done with the movie on tuesday we are going to do some kind of practice tests or something? idk. But hopefully i remember the things that we learned in the beginning of the school year. I don't really know what else to say.. but 3 day week this week!! Well at least for softball players:)

Reflection for 2/28

blah, idk what to do for this blog.......there's really nothin to put bc we just read the book all week.......

hrmm............there really is nothin.........i guess i could.....dude! there's nothin that i can even remember that has anything to do with math atm...........blah

that and flatland......that book confused me.....a lot........i really didnt understand too much of it.....if someone could help me out with that, thatd b great

thats all i got......

:/

or i guess i could put the trig chart.......but that wont make much sense on here......but o well

0º sin0=0 cos0=1 tan0=0 cot0=Ø sec0=1 csc0=Ø
30º sin30=1/2 cos30=√3/2 tan30=√3/3 cot30=√3 sec30=(2√3)/3 csc30=2
45º sin45=√2/2 cos45=√2/2 tan45=1 cot45=1 sec45=√2 csc45=√2
60º sin60=√3/2 cos60=1/2 tan60=√3 cot60=√3/3 sec60=2 csc60=(2√3)/3
90º sin90=1 cos90=0 tan90=Ø cot90=0 sec90=Ø csc90=1


i guess thats all....just need a little help with the book though.........
:/

Reflection

This week went by slow, i guess it was because we had to read that book. I'll talk about the stuff from Chapter 10. In chapter 10 we learned, some more stuff dealing with trig stuff and the trig chart. We took 2 quizzes when we began learning this chapter, the first one we took i thought i did terrible on but did decent. Throughout the week, we ended up getting through the section 10-4. We learned a lot of formulas we have to memorize like: sum and differences with sin and cos, half and double angles, and many more. I found the easiest formula to work with was tan alpha + tan beta/1-tan alpha tan beta. I also found the sin(alpha + beta) was easy. But overall, this week was a simple week with mostly easy stuff we learned, just had to memorize a lot of formulas. And if anyone wants to help me with what we learned on friday, that would be great because i dont totally understand it.

REFLECTION #28

Welllllll it's the weekend again so here's another blog. In this one and in some more in the future I think I'm just going to go over random concepts in trigonometry. Better start reviewing that now before the big trig exam.

So here's a few random examples:

Ex. 1.) Prove that sin(x + pi) = -sin x
*Okay first looking at this problem you know that you have to use the sum formula for sine to expand this problem out
**when they say "prove that", all that means is just to work the problem and see if you get the same answer as they give you
*So you'll use this formula: sin(a + b) = sinacosb + cosasinb (*where a is alpha and b is beta)
*and then all you do is plug x in for alpha and pi in for beta and simplify the expression
*So you get sinxcospi + cosxsinpi (*Note* (sin x)(cos pi), just in case it just looked like a lot of letters..)
*and you can simplify cos pi to give you -1, and sin pi is zero
*So plugging that in you get>> sin x(-1) + cos x(0)
*The cos x gets canceled out and you're left with -sin x.

Ex. 2.) If tan a = 4/3 & tan b = 12/5 ..find cos (a - b) if 0(*again, where a is alpha and b is beta)
*Sooo they tell you to find cos(a - b) which means you're going to have to use this formula:
cos(a -b) = cosacosb + sinasinb
*You're going to have to draw triangles for this one so you can get sine and cosine for the formula..since you're just given tangent
*So first you draw your "alpha" triangle in the first quadrant. *you know it's the first quadrant because in the directions they tell you alpha and beta are both between zero and 90 degrees, which is in the first quadrant.
*For the alpha triangle you use tan a given to you in the directions which is 4/3. and the hypotenuse is 5 because it's a perfect 3 4 5 triangle.
*So you need to find sin a and cos a for that triangle
*you get that sin a = 4/5 and cos a = 3/5
*Now you draw your "beta triangle and you will use tan b = 12/5. and your hypotenuse is 13 because it is also a perfect triangle
*then you get that sin b = 12/13 and cos b = 5/13
*Now all you do is plug in all these numbers into the formula for cos(a - b)
*So you get (3/5)(5/13) + (4/5)(12/13) and simplifying that you get 63/65

Ex. 3.) If tan a = 2 and tan b = 1 ...find tan(a + b)
*for this one you'll have to use the tangent sum formula>> tan (a + b) = tan a + tan b/1-tanatanb
*Since you're given tan a and tan b all you have to do is plug in the numbers you're given into the formula
*So you get 2 + 1/1 - (2)(1)
= 3/-1
So you're answer is -3

Enjoyed you're trig review for the day?....hahaa yeah me neither