Law of Sines/Cosines

Mary Christmas :)!!!!!

Law of sine is used when given at least 2 sides and one angle, or given 2 angles and one side. Law of cosine is used when given three sides, or three angles, or one angle and two sides. When we solve a problem, we better just use law of cosine one. And continue the problem using law of sines. Because the law of cosine is more complicated than law of sine, we can avoid unassary calculation errors by choosing a simpler method. Law of sines is that the fraction of one side and it's opposite angle is proportional to the other set of angle and side.  Law of cosine is a^2=b^2+c^2-2bc CosA.  And same for solving other sides. 

Chapter 4 Summary

In the beginning of chapter 4, we talked about angle measurements. We learned how to convert between degrees, minutes and seconds. We reviewd about complementary and supplementary angles. We learned about unit circle. And trigonometric identities, sine, cosine and tangent. We learned how to graph them.  We also learned special angles' values.  In chapter4.5, we learned about verifying trig identities.  And then we learned how to use identities to solve trig equations.  

Mr. Unit Circle.

The unit circle has a radius of 1. Because the radius is one, we can measure sine,cosine, and tangent. The Pythagoras theorem is a^2+b^2=c^2, it only works for right triangles.  When we solve trig problems, unit circle is extremely helpful. It saves up a lot of time. Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. I like unit circles because they make things easy. 

Trigonometric Equations

To solve trigonometric equations: 1. combine all th elike terms. 2. collect trig terms on one side and constant on one side. 3. apply a trig identity if applicable. 4. factor if possible. 5.isolate trig function. 6. solve for variable.  i dont like solving trig equation, because they are so hard, and they hurt my brain,,,,,,,,

Verifying Trig Identities

when we try to verify a trig identities, we first simplify the more complicated side. 2. find the common denomenators. 3. change all trig functions in terms of sin and cos. 4. use an identity to verify.  when we are stuck at certain point, we should try to use another way to approach the problem rather than keeping struggling.there are various ways to verify trig functions.i like doing those problems, because they are torturing my mind.

tangent

tangent has a function y=Atan(Bx+c)+D. tangents are undefined at 90 and 270 degree. has asymptotes at pi/2 and 2pi/2. the period of tangent graph is pi/B. cotangent occurs at asympyotes of tangent graph. i dont understand how to calculate the vertical asymptotes of tangent graphs.

Sine and cosine

Sine is using the opposite side dived by the hypotenuse. Cosine is using the adjacent side divided by the hypotenuse.  When we solve a triangle, it is really helpful to use sine cosine and tangent.  When we draw a graph for sine and cosine, the basic equation is y=ASine or Cosine (Bx+C) 

Where A is the amplitude of the graph. To get the period of the graph, we use the formula 2pi/B.  If we need to know the phase shift, we set Bx+C equals to zero and solve for x.  It is the same for both sine and cosine graph.   

chapter 3 summary

In chapter three, we learned about rational zeros, rational functions, divisions and asymtopes.  The chapter was not extremely hard to me, it is just a little tidious when doing some steps.  When we are given a function and a zero or factor, we use division to find the other zeros. The power of the first x term indicates the number of zeros.  When the problem asks me to complete the factorization, I divide all the given zeros or factors to get the equation into x square form. We need to put the answer into the form of q(x)•d(x)+r(x).  I liked this chapter eventhough I think I did bad on the test.  

Rational functions

Rational functions are a polynomial over a polynomial. It looks really complicated,but when we take it down into pieces, it is actually easy to graph.   In order to graph the functions, we need to find three things. First, the horizontal,slant and vertical asymtopes, secondly, holes, if the graph has any. To find the vertical asymtopes, we set the denominator to zero and solve for x. To find the horizontal one, we have to test if the value of n is larger than m, if it is, there's no horizontal asymtopes. If n<m! the answer would be y=an/bm. If they are equal, the answer is y=0.  We divid the functions using long division I order to find the slant asymtopes. It only exists if n>m.  We factor and cancel to find the holes, and the answer should be a ordered pair. 

zeros of function

in order to find the zeros of  function, we can use long divition and find the complete factorization. We use the given fator to find the complte factorization. fist we have to test weather or not the given number is a factor. After we find the complete factorization, we can use the quadratic formula to find the zeros. If the answer involves with a negative square root, we have to use imagery numbers to express the answer. I think the problems with the imagery numbers are hard for me, it is really easy to make mistakes during the calculations. I did not understand the problems which they give you multiplicities and zeros, and ask me to find the function.

Piecewise functions

Piece wise function is a function composed with different pieces of different functions. The piece wise function could be continuous or discontinuous. Continuous means that there is no >or <, no gaps in the graph! and can be drawn without lifting the pencil. When graphing a piecewise function, we draw a open circle for >and<, we draw a solid dot for less than or equal to and greater than or equal to. When we have a greatest integer function, expressed with [[x]], the function goes on the same wit a pattern.  The piecewise functions with absolute values flips the results and it looks weird.... This chapter is kind of hard but fun. 

The F(x) man

The project was using the different equations and translating them in order to "kill" all he bad guy.The project was really fun! The reflection section was hard for me. My favorite superhero isCaptain Abs, because he kills a lot of targets at one shot.  The project helped me to practicetranslating equations. When I was doing the reflection part I forgot to put the - sign outside of the equation several times. The equation was wrong when I did it, but I fixed it later. 

What is a function?

A function is written n the form of f(x) equals to a function. f(x) is equivalent to y. In order to test rather the equation is a function, we can use the vertical line test.  It is only a function when the one x glue has one y value. X-intercepts are also called solutions and zeros. In order to find the x-intercepts, we set the f(x) to 0 and solve for x.  When we can't factor a quadratic equation, we use the quadratic formula. Functions and graphs are not so hard, I like them. 

What I learned this week.

This week we reviewed real numbers, inequalities, absolute values, and equations of circles. In addition to inequalities, we learned how to write set notations, and how to use sign chart method to solve inequality equations with x^2. We learned how to solve inequalities with absolute values. We also reviwed the distance formula and midpoint formula. We reviewed intercepts and we learned the formula for semicircles. We reviewed how to rewrite the equation of circle if it is not written in the standard format. We also reviwed how to graph a circle.  

All about me

My name is blair.. :) I like music, I play guitar, piano, and I write songs.  I like to read the lyrics when I listen to music. My favorite subjects in school are English, music, math and science.  I do not like history.  I like to travel, I prefer tropical areas but not the near beach. I like humid air but not salty humid air.  I love animals, and I wish to have a cat named Peanutbutter in the future. I had a beta fish named Peanut, I loved it a lot.  He died last year and I was really sad. :(    I don't do sports, I am a more musical person.  I came to America 3 years ago, I am enjoying the experience so far. I wan to study music in college. My dream schools are USC Thornton College of music, and Berklee college of music.   My goal this year is to get solid grades and aim for 800 on SAT in Math section.  I guess that's all about me, this thing is so long ...... :D