Mr. Unit Circle

The unit circle is very useful in many trigonometric equations. The unit circle is set on a graph with the main points being at 0, 90, 180, 270, and 360 degrees. In between these angles are several smaller angles such as 45 degrees and 60 degrees and all of these have a radian measure and points on them. The points on the angles correspond to their sine and cosine with sine equalling the y value and x equalling the cosine value. Tangent can also be represented in these points through sine/cosine or y/x. Here is a picture of a unit circle. 

Law of Sines/Cosines

The law of sines and cosines are two ways to solve triangles when they are not right triangles. The law of sines can be used when you have two sides and an opposite angle or two angles and an opposite side. The equation is sinA/a=sinB/b=sinC/c with the capital letters equaling the angles and the lowercase letters equaling the side lengths. The law of cosines can be used when you have 2 sides and an included angle or three sides and no angles. The equations for law of cosines are a^2=b^2+c^2-2(b)(c)(cosA), b^2=a^2+c^2-2(a)(c)(cosB), c^2=b^2+a^2-2(b)(a)(cosC). 

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. 

Rational Functions

A rational function is a polynomial divided by a polynomial, basically a fraction with polynomials in the numerator and denominator.  Examples are x^2+3x-2/3x^2+8x-7 and 2/x-7 or even 3x^2-5. However rational functions are not allowed to have square roots or fractions in them like 3-sqrt 4/6+x. Rational functions can generally be expressed as f(x)= p(x)/q(x) as long as q(x) does not equal zero.  To find the roots of a rational function you have to find where the graph crosses the x axis or the x intercepts of the equation. To find the vertical asymptotes you just set the denominator equal to zero and solve for x. For horizontal asymptotes if the degree of the numerator equals the denominator then you divide the leading degree of the numerator and denominator, if the degree of the numerator is less than the denominator than y=0 and if the degree of the numerator is greater than the degree of the denominator then the horizontal asymptotes do not exist. Here is an example of a rational function graphed.

Finding Zeros of a Function

The zeroes of functions is where the graph crosses x. To find zeroes of a function you set the equation equal to zero. Another way to do/say this is to set y equal to zero. Once you do this simply solve the equation for x. Some times you will get more than one answer and that's okay. When you get the same number multiple times that number has multiplicity. For example if you get the number two three times you would say 2 has a multiplicity of three and if 5 appeared seven times you would say five has a multiplicity of seven. Here is an example of finding zeroes. 

Piece Wise Functions

A piece wise function is when a function has two different formulas that are both defined on the domain of f. Even though they are two different equations they are looked at as one function. They are usually used to represent the negative and the positive numbers. For example one equation usually has the restriction of x<0 and the other equation is x>0. To graph a piecewise defined function you graph each equation as you would normally and then it is either considered continuous or discontinuous. A continuous graph is when you don't have to lift your pencil when tracing the graph. A discontinuous graph is disconnected in places. Here is an example of a piecewise function problem.

Superhero Transformations

Today I worked on superhero transformations. There are six superheroes, lady straightedge, pawabawa, robo-grow, captain abs, radical girl, and bipolar tommy. Lady straightedge represents a straight line, her equation is f(x) = x and she has nuclear fusion blasts from her eyes. Pawabawa is a parabola his equation is f(x) = x^2 and he has parabolic kinetic rays. Robo-grow is from the exponential family, her equation f(x) = 2^x and she has a ginormous robot suit. Captain abs represents absolute value his equation is f(x) = |x| And he has finger beams. Radical girl represents a square root her equation is f(x) = sqrt x and he has ninja skills with a sword. Lastly bipolar tommy represents Cubing and his equation is f(x) = x^3 and he has a curvy laser gun. All these superheroes and their powers helped to complete the missions. 

What is a Function

A function is usually represented by the letter f. In a equation it is usually written like f(x) (meaning f of x). This f(x) simply represents why in an equation. For example you can rewrite f(x)=3x-5 as y=3x-5. To use functions in equations, take whatever's in the x spot and replace it with the x's in the equation. For example for f(2)=3x-5 replace the x with a 2 like 3(2)-5 and the final answer is 1. You can also use functions in domains and intercepts. Here is another example of a function problem.

What I Learned This Week

This week I learned lots of fun things about math. First I learned about inequalities and set notations.  We learned how to express inequalities and write them in interval notation.  Also I learned to use the Sign Chart Method which is

1. Make the inequality with a zero on one side
2. Factor
3. Mark where factors are zero
4. In each interval determine the sign

I also learned about absolute value and its properties and I learned about the rectangular coordinate system and how to graph with inequalities. Lastly I I learned and circles and graphs, the equation of a circle, semi circles and symmetry. Here is a picture of my notes....

All About Me

My name is Mary, i am 16 and I am bad at math. 

Some interesting facts about me are...

• i love sourcream, reading, and boybands (its only two don't judge please)
• i have a strange fascination for World War II
• i love shopping for school supplies more than clothes
• Costco will always be more precious to me than Forever 21 or any other store....at all
• i have 7 siblings, 3 brothers and 4 sisters
• i was home schooled until high school
• i currently have a minor slash major obsession with Bastille, Megan Trainor, Ed Sheeran and TRXYE
• there's nothing i want more than a unicorn pillow pet
• i was cher lloyd in another life
• and i really really don't like math but i'm trying to!!

aaaand yeah!

thats me....

yay

okay

bye