Polar coordinates

Polar coordinated are coordinates plotted on a circle graph. The circle graph is basically an extension of the unit circle. Since it is an extension of a unit circle, there are two kinds of points that one can plot: rectangular and polar. The rectangular coordinate points are the basic (x,y) coordinate pair where x=rcos(theta) and y=rsin(theta). The second is the polar coordinate which is represented by the coordinate pair (r,theta) where r2=x2+y2 and theta is equal to the inverse tangent of theta. There are four ways in which a polar coordinate pair can be written: (+,+)(+,-)(-+)(-,-). To plot polar coordinates. On has to find the value of theta on the unit circle then either r circles out wards or r circles towards and last the origin in the opposite way if it is negative. 

Second Semester Summary

Second semester was a little better than first semester in terms of being organized, but that quickly failed and I regressed to my original slob form. There is only two weeks of school left and I am determined to get them over with as soon as possible because I am so done with school. I have three binders that I carry around me now and days because I stuff papers and such into random binders and so I have to carry them all just in case I need one specific worksheet I need to turn in. Math wise, I am not very happy with myself in how I did second semester. I began a lot better than first, but my test scores were constantly low and blogs were forgotten about constantly as well. These two things are going to take a huge toll on my grade and that is one thing that I am very scared of. In terms of my other classes like I'm just done.

Trig Review Week

The presentations that my classmates gave were extremely helpful in remembering certain topics and methods of solving a problem. Many of the concepts I had forgotten that we had even learned therefore it was really nice to be able to refresh my memory in such a small amount of time. I think the actual trig itself is the hardest since there are so many of the identities to memorize and I get overwhelmed. I need to review a lot for the final since many of the topics that we talking about during the presentations were somewhat difficult for me as well as the induction/deduction arguments.

Repeating Decimals

Every repeating decimal is the sum of an infinite geometric series. In order to find the quotient of integers (the fraction) of a repeating decimal, first, the decimal has to be written as a quotient of integers and the geometric series has to be seen. From the broken up decimals, the sum should be really easy to find. After finding the sum, the corresponding parts of the equation S=a/1-r. The final answer should always be checked by a calculator just in case.

Parametric Equations

In parametric equations, first, a t/x/y has to be made in order to be able to graph the equation. The graph should have arrows pointing in the direction the line is going. After sketching a graph, one has to eliminate the parameter using algebra and trig identities in order to solve a equation. The parameter can be eliminated through substation or elimination, so long as the final answer is a rectangular equation.

Partial Fractions

Partial fractions are basically the reverse of multiplying two factors out into one. Given a linear equation, the denominator has to be factored out into its two (or more) factors. Above every factor should be a letter so the set up looks something like A/(x-2) + B/(x+3). However, if one has a denominator that cannot be factored, a quadratic equation, then the numerator becomes Ax+B (and the variation of this set up). If the denominator is squared at one point, when factoring this out, this one equation that is squared has to create two different partial fractions. One must be the factor in the parenthesis and the other is the whole factor (A/x+B)^2 + B/x+B.

Power of Hanoi Mathematical Induction

Tower of Hanoi is a game in which there are disks of different sizes stacked onto of eachtoher in decreasing order "( smallest on top and largest on the bottom) and the whle point of the game s to move the disks across to another empty disk holder in the least amount of moves. For the power of hanoi, the mathematical induction began with 2^n, the number of moves it took to the initial tester to get the equation 2^n-1. This is then proven by plugging the variables into the induction equations and then solving to ring true.

Sequeneces and Series

A sequence is an ordered collection of objects where repititon is allowed. The general form of  a

sequence is A1, A2, A3...An, An+1. There is the arithmetic sequence and the geometric sequence. A Series is the sun of the terms in a sequence. For this one as well, there is an arithmetic describes and a geometric series. Both series and sequences are used to prove or disprove patterns.

Graphing systems of Inequalities

There are three different graphs that one could stumble across when dealing with systems of inequalities. The equation of a line is y=mx+b, a parabola y=(x-h)^2+k, and the circle x^2+y=r^2. From one of these equations then, you have to pick a test point. Finally, you have to shade the plane containing the test point. If the test point satisfies the equations. If the sign of the inequality equation is greater than or equal to you, then the line of the graph would be solide. However, if the equation is less than one, then the line of the graph will be dotted. 

Cramer's Rule

Cramer's Rule is used for a system of linear equations that has as many equations as unknowns and is valid whenever the system has a unique solution. The main equations are x=DX/D and y=DY/D. The way the actual solving process is set up is you put only the coefficients of x and y then cross multiplying as you do in matrices in order to get an answer. Then for the second equation the x values would be replaced by the value of the sum and the same for the y too. The point of Cramer's Rule is to not need more than one variable in order to solve a system.

Systems of equations

There are two kinds of systems of equations, the inconsistent and consistent. If the system is inconsistent, then there is no solution and there are parallel lines. However, consistent mens that there is one solution which is called the independent and infinite solutions (dependent).

There are two methods in which systems of equations can be solved. The first is substitution and in substitution, you have t o first solve 1 equation for one variable then substitute it into another then solve for the variable. From this point you would have to back substitute in order to find the 2nd variable. The second method is elimination where you would interchange any 2 equations then multiply by a constant number then add one equation to the other to eliminate a variable. 

Graphs of Polar Equations

So when graphing polar equations, always begin by creating a Table-of-Values. If r=acos@, then the circle is along the x-axis with diameter "a". If r=asin@, then they exist with a diameter and if r=a then the circle is at (0,0) with radius "a". There are a couple different kinds of polar equations. First are Limacon which are basic circles that has the chess  part. The second are Cardiods, which are special cases of limacons. Lastly, rose curves and lemisactes are also polar equations.

Polar Coordinates

Polar Coordinates are basically the position of a point in a plane, the first being the length of a straight line represented by "r" which is connects the point to the origin, and the second angle "theta" (represented as @ in this blog post due to the fact I do not have a theta sign) is made by this line with a fixed line. There are four ways to get to the same point on a polar coordinate: (r,@), (-r,@+pi), (r,-2pi-@), (-r, -@-pi). In order to convert between rectangular (x,y) and polar (r,@), remember that the two are equal to each other therefore can be used in stead for each other. Remember these two key equations!

r= /x^2+y^2    (the square root of x squared plus y squared

and 

@=tan^-1(y/x)   (the inverse tan of y divided by x.

Rotating Conic Sections

In order to rotate a conic section, one must begin at the starting point which has the equation: Ax^2+Bxy+Cy^2+Dx+Ey+F=0.

The first step is to find the angle through the equation cot20=(A-C)/B where 0<theta<90.

The second step would be to replace the x's and y's in order to get x=x*cos(theta)-y*sin(theta) and y=x*sin(theta)+y*cos(theta).

The last step would be to use algebra to simplify the equation. You should know your basic trig functions in order to aid in the process of simplifying.

Parabolas

Parabolas are one section of Conic Sections. It is comprised of two equations:
1. Standard Form:
a. (x-h)^2=4c(y-k) and
b. (y-k)^2=4c(y-h)
2. General Form: y=ax^2+bx+c
The standard form "a" creates a parabola that opens up or down whereas the standard form "b" creates a parabola that opens left or right. The side in which the parabola opens is determined by whether or not the initial equation has a positive or negative sign. If it is positive, the graph will open to up or to the right whereas if it is negative, the graph will open down or to the left. The general form can always be converted back to the standard form through simple algebraic steps.