Chapter 4 Overview

In chapter 4 we learned a lot about trigonometry and its functions. In 4.1 we reviewed the basic ways to measure angles from a graph and learned about things like the standard position. In 4.2 and 4.3 we reviewed the sine and cosine functions and also the graphs of both of these and in 4.4 we learned about cosecant, cotangent and secant. 4.5 was about verifying trig identities which is basically getting one side of an equation to look like the other side through trig identities. In 4.6 we learned about sum and difference formulas which involved about thirteen formulas that we had to memorize. 4.7-4.9 were all about how to solve trig equations and also simple harmonic motion. Here is a list of some of
the identities we had to memorize. 

Trig Equations

Trig equations are used with SOH-CAH-TOA. There are six main trig functions that exist which are: sine=opposite/hypotenuse, tangent=opposite/adjacent, cosine=adjacent/hypotenuse. The last three trig functions are the reciprocal of the first three with secant corresponding to cosine, cotangent corresponding to tangent and cosecant corresponding to sine. You can use these equations to find the angles of right triangles. Another way to write the reciprocal functions is: cotangent=1/tan, cosecant=1/sine and secant=1/cosine. Here is an example of the trig functions. 

Verifying Identities

This week we learned about verifiying identities. Verifying identities means making one side of an equation look like the other side. There are five tips on verifying and one of them is to simplify the more complicated side (for example simplify the addition side not the multiplication side). The second tip is to find common denominators in fractions. The third tip is to change the functions in terms of sine and cosine. The fourth and fifth tips are to use an identity and to multiply by a conjugate pair. If you use all these tips and use the identities correctly you should be able to verify identities.  Here is a picture of some of the identities that are essential to verifying. 

The Tangent Graph

Tangent is another function that is similar to the ways that sine and cosine work but the graph is much different. Unlike sine and cosine, tangent requires asymptotes which show where the graph cannot cross. Also each period of a tangent graph is not connected but are there own separate graphs that go up and down infinity. The tangent equation is y=Atan(Bx+C) + D. To find asymptotes of a tangent graph use the equation pi/2+npi. The equation to find the period is pi/b and to find the x intercepts is Bx+C=npi and A is the amplitude. Here is a picture of a tangent graph.  

Sine and Cosine Functions

Sine and cosine are functions that are used to find angles and sides of triangles. On a graph cos=x and sin=y. To use sin and cosine you use the acronym SOH CAH: soh means sin equals the Opposite side from the angle over the hypotenuse of the triangle. CAH means cosine equals the adjacent side of the triangle over the hypotenuse of the triangle.  Secant and cosecant are two more functions that relate to cosine and sine.  Secant equals 1/cos and cosecant equals 1/sin. Basically they are the reciprocal of cosine and sine. Here is a picture showing how to use sine and cosine in a triangle.  

Summary of Chapter 3

The main focus of chapter 3 was to find the zeroes of a function and there are many steps and different ways to do that. First we learned about polynomial functions and looked at end behavior and how to tell if a function is even or odd. We also learned about multiplicity which tells how many zeros a function has. We also learned how to do division of a polynomial function using long division and synthetic division which can be used to find the zeroes of a function. Another way to find zeroes is using p/s where p equals the factors of the constant over the factors of the leading coefficient and then you find all possible zeros and test those zeros in the equation. We also learned about approximating zeros which is basically just dividing the interval in half until you get a zero. The last thing we learned is rational functions and we learned how to find vertical, horizontal, and slant asymptotes and the holes of an equation. Here's a picture showing how to find all these asymptotes and holes.