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. 

Trigonometric equation

A trigonometric equation is any equation that contains a trigonometric function. In order to solve a trig equation, we have to use both the reference angle we have memorized and a lot of algebra we have learned. For example, solve sin(x)+2=3 for 0 degree < x< 360 degree. I'll first isolate the variable containing term. Sin(x)+2=3 sin(x)=1. Now I'll use the reference angle I have memorized x=90.

Verifying identities

We have different identities to deal with the trig formulas, and those identities makes it easier to solve. For example, sin^2x+cos^2x=1, and 1+tan^2x=sec^2x. In order to solve the problem completely, we should use some rules, first simplify more complicated side. Second, we should find their common denominators if it is possible. We should always change all trig function in terms of sine and cosine. And finally, we can try to plug them into the identities we learned this week. 

Tangent functions and graph

We learned how to solve and graph tangent equation. The tangent will be undefined wherever it's denominator is zero. The tangent wil be zero wherever it's numerator is zero. Therefore, the tangent will have vertical asymptotes wherever the cosine is zero. One basic knowledge about tangent is that it equals to sin/cosine. 

Chapter 3 review

We basically learned how to deal with polynomial function, long division, synthetic division, rational zero test, finding approximating zeros, and rational functions. For polynomial function, we learned that the a i are real numbers and are called coefficients, also the term an is assumed to be nonzero and is called the leading term. Then we learned how to use long division and synthetic division and we can use both way to divide functions. For approximating zeros, we divide the interval in half to find its midpoint and complete f(m).

Sine and cosine functions

We learned how to deal with sine and cosine fuction this week. The sine and cosine are the two most prominent trigonometric functions. All of other trig functions can be expressed in terms of them. The sine theta is the vertica coordinate of the arc endpoint, and the cosine theta is the horizontal coordinate. 

10/3 What I learned this week: rational function

This week we learned "real zeros of polynomial functions", "approximating real zeros", and "rational functions". Among those new materials, the rational function is the most memorable and abstruse to me. The rational function is a function that one is divided by the other to form a ratio of two polynomials. In order to graph a rational function, it requires the vertical and horizontal asymptotes. For vertical asymptotes, just simply set the denominator of the function equal to zero and solve. For the horizontal asymptotes, it is more complicated. You have to find n^th and m^th degree of the polynomial and according to different conditions, the answer varies.