Systems of equations.

Hello,

Today I will be covering systems of equations. Systems of equations are a collection of multiple equations with solutions. These systems can be consistent or inconsistent, and if they are consistent it means that they have a solution, and if they are inconsistent, it means that they have no solution. If they are consistent, they have the option be be either dependent or independent. If they are dependent, it means that the solutions are zero, and you must solve for z and substitute z for t. Independent solutions mean that they are real numbers greater or less than 0. We can solve systems of equations using elimination or substitution or even matrices.

Thanks,
Kaili Chiu

Graphs of Polar Equations

Hi,

Graphing Polar Equations is quite a task. We have to first master polar coordinates, which was explained in the previous blog post. Now that we've reviewed that, we can take a look at the graphs. We use the conversions of r^2 = x^2 + y^2. and x=rcos(theta) and y=rsin(theta). A circle could be r=asin(theta). There are limacons, cardioids, rose curves, and lemniscates. We played around with them in class, when we made art to create a picture. Overall it was a pretty cool lesson, and I enjoyed it a lot.

Thanks,
Kaili Chiu

Polar Coordinates

Hello,

Today, I will talk about Polar Coordinates. Polar coordinates are on a different plane than the (x,y) coordinates. (x,y) coordinates are on the Cartesian plane, whereas the Polar coordinates are on a polar plane. Instead of the common (x,y) point, on the polar plane it is (r,θ). θ is the angle at which the axis is rotated in relation to the positive x-axis of the cartesian plane. Although this seems very confusing at first, it is only just tedious. All in all, though intimidating, this lesson was quite easy.

Thanks,
Kaili Chiu

Rotating Conic Sections

Hello,

Conic sections can be rotated around the origin, substituting x prime and y prime for x and y. We can use unit circle measurements to find the angle. Using the unit circle measurements for the angle, we can adjust the entire cartesian plane to rotate the graph. Some useful information on rotating conic sections include knowing basic trigonometry such as x=xprimecos(theta)-yprimesin(theta), and the y point. These allow the student to plug x and y points into the original formula given. The result will be the rotated equation. Simplify and turn in your answer!

Thanks,
Kaili Chiu

Parabolas

Hello,

Today I am going to talk about parabolas. They are quadratic graphs and are shaped like a "u", although if the standard form is negative, it is shaped like an "n". The a value in standard form determines the opening up or down, and if a>0, then it opens up, and if a<0, it opens down. Parabolas have a vertex, a focus, and a directrix. On a regular (x-h)^2 = 4c(y-k) graph, (h,k) is the vertex, (h, k+c) is the focus, and h-c is the directrix. The axis of symmetry is on the same x coordinate as the vertex of a standard parent graph. Parabolas are present in many fields, including physics and engineering.

Thanks,
Kaili Chiu

Semester 2 Goals

Hi,

There are some things I did well in last semester. These were memorizing the trig identities, and trying to keep my grade up in the end of the semester. In addition, towards the end of the semester, I started to really focus on how the pieces in the sections of the chapter fit together to apply concepts. however, there are inevitably some things I did poorly in. Some of these included forgetting to finish homework, forgetting to take notes in class, and procrastinating. I would really like to fix these problems, and the only way I can do that is to put in hard work. During Christmas break, there wasn't anything really funny, but the only thing that could be funny is that my cousin came over and accidentally went to the neighbors house.

Law of Sines and Cosines

The law of sine and cosine compares parts of a triangle. There are many formulas for the law of cosines, however the law of sines is a ratio, which compares sin(X)/x = sin(Y)/y. The law of cosines is more complicated.  The formulas for law of cosine are all variations of c^2 = a^2 + b^2 - 2abcos(C).
It is basically the pythagorean theorem, but it includes a value for an angle, which has the - 2abcos(C).
The other sides of a and b just switch around the variables. Both these laws are widely used in life appliactions, as they do not need to account for right triangles.