Rotating Conic Section

We learn rating conic section on chapter7.4. We studied conic sections with equations of the form: Ax^2+Bxy+Cy^2+Dx+Ey+F=0. Step 1: find angle cot2pi=(A-C)/B where pi is greater than 0 but smaller than 90 degree. Step 2: replace x and y: x=xprime*cosine-data-yprime*sine-data. y=xprime sine-data+yprime cosine data. Step 3:plug these into the original equation and simplify to solve. There are three constant equation help us to recognize different graph shape. Parabola:B^2-4AC=0.

Ellipse: B^2-4ac smaller than 0. Hyperbola: B^2-4AC greater than 0.

Parabolas

We did parabolas on chapter 7 of this semester. Through researching on internet, I now can identify "parabolas" as the curve formed from all the points (x, y) that are equidistant from the directrix and the focus. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola up the middle) is called the "axis of symmetry". I found this chapter to be relatively interesting since I personally like the graphing section of math a lot. A parabola equation is (x-h)^2=4c(y-k) for vertical shift. (Y-k)^2=(x-h) applies for horizontal shifts. 

My second semester goal

My second semester's goal for this class is that I wish I can be more focus at the lecture contents. I'll do detail review before every quiz or test in order to get a descent grade. I hope I can enjoy the math more and explore more math theories in mathland. I was sometime absent-minded last semester and now looking forward to improve that. For Christmas, my family went on a trip to LAke Tahoe. They had beautiful views but the weather was freezing. In general, my break was quiet peaceful and relaxing. 

My polar equation graph

Chapter 4 Summary

Chapter 4 is composed of 11 sections. All of them are very challenging concepts to be absorbed. We learned the trig identities of sine, cosine, and tangent; the trig graph; verifying trig identities; inverse trig functions; harmonic motion. Among these various lesson, I think I am pretty confident on 4.8: Inverse Trigonometric Functions. From researching the definition, I found out that In mathmatic, the inverse trigonometric functions (occasionally called cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). For example, a question is asked to find the inverse function of sine that's equal to 1/2. The answer would be pi/6. We could answer this in a complete sentence "the angle whose sine is 1/2 is pi/6". The algebraic abbreviation for that sentence is arcsine=pi/6.

Mr.Unit Circle

The "Unit Circle" is a circle with a radius of 1. Frequently, especially in trignometry, the unit circle is the circle of radius one centered at the origin (0, 0). Being simple, it is a way to learn and talk about lengths and angles. The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement here. Because the radius is 1, we can directly measure sine, cosine, and tangent.
For example:
What happens when the angle, θ, is 0°?

• cos 0° = 1, sin 0° = 0 and tan 0° = 0

What happens when θ is 90°?

• cos 90° = 0, sin 90° = 1 and tan 90° is undefined

Law of sines Law of cosines

This week we learned the law of sines and the law of cosines. In trigonometric, the law of sines is an equation relating the lengths of the sides of any shaped triangle to the sines to its angles. According to the law, a/sine A=b/sine B=c/sine C, where a,b, and c are the lengths of the sides of a triangle, and A, B, C are the opposite angles. Law of cosines is an equation relates the lengths of the sides of a triangle to the cosine of one of its angles. The law of cosines generalize the Pythagorean theorem, which holds only for right angles.