Law of Sines and Cosines #15

To use the law of sines, a triangle must either have 2 sides and an opposite angle or 2 angles and an opposite side. The equations to use are sinA/a = sinB/b = sinC/c. When approached with the situation of sin being attached to a letter, take the inverse of both sides. A sin and its number are not allowed to be separated because they are one pair and can only be apart it the inverse is taken. The law of cosines is only used once, then use the law of sines. To use this law, a triangle must be given 2 sides and an included x or 3 sides. The formulas for the law of cosines are a^2=b^2+c^2-2bccosA, b^2=a^2+c^2-2accosA and c^2=a^2+b^2-2abcosC. 

Chapter 4 Summary #14

Chapter 4 was a difficult one. First, we learned about angles and their measurements. Standard position is when the initial side is in the positive x-axis. Coterminous angles are angles that have the same terminal side. For complementary angles, add 90 degrees or pi/2. For supplementary angles, add 180 degrees or pi. The arc length formula us s=rtheta. Theta must be in radians and there are no exceptions! Then, we learned about special triangles: 45. 45, 90 degrees and 30, 60, 90 degrees. Then, we went into verifying trig identities. Then, the sum and difference, double, and half angle equations, which I found to be the most difficult. Inverse trig functions were relatively simpler for me. I learned to always unrationalize the arc length. 

Mr. Unit Circle #13

The unit circle is extremely helpful. It provides the cosine and sine values of the important degrees. It also gives the degrees in radians. The unit circle is a circle with a radius of 1. Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. There are also trig functions on the unit circle.

Trig Equations #12

There are many different trig equations. An example of a sum and difference equation would be cos105 degrees. You must add two degrees together to make 105. Use the equation cos(x+y)=cosxcosy-sinxsiny. Cos(60+45) equals to 105 degrees, so now, you must find the sine and cosine of 60 and 45 degrees. Then, just plug them into the equation. An example of a double angle would be sinx=-1/4, solve for cos2x and sin2x. First, you would use the identity cos2x=cos^2x - sin^2x. In order to find cosx, use the identity sin^2x + sin^2x = 1. Since sin is already given, plug it in and solve for cos. 

Verifying Trig Identities #11

The first suggestion in verifying a trig identity is to simplify the more complicated side. Then, find common denominators. After that, change all the trig functions in terms of sine and cosine. Finally, use an identity! The goal in verifying trig identities is to make it equal to the other side of the equation. If you do not get it the first time, do not be discouraged and try it again. Using identities is the most important suggestion. 

Tangent #10

Tangent of theta is equal to the y value (cosine) / the x value (sine). A way to find tangent is to remember TOA. The TOA represents that tangent = opposite/adjacent. The tangent is always positive in the first and third quadrants. Tangent graphs are undefined at 90 degrees and 270 degrees and the asymptotes are at pi/2 + kpi. The periods are shorter that the sine and cosine graphs. The x intercepts equal npi. The period is pi/B. 

Sine and Cosine #9

Sine of theta is equal to the y value of a triangle. Cosine of theta is equal to the x value of a triangle. A way to find sine and cosine is to remember SOH CAH. The SOH represents sine = opposite/the adjacent side of the theta. The CAH represents cosine = adjacent/the hypotenuse of the triangle. Sine is always positive in the first and second quadrants. Cosine is always positive in the first and fourth quadrants. 

Chapter 3 Summary #8

Chapter 3 covered polynomial functions, multiplicity, synthetic and long division, the rational zero test, approximating zeros, and rational functions. The equation for a polynomial function is: AnX^n + An-1X^n-1 + An-2 + A1X^1 + Ao. A is the coefficient and ^n is the degree. An example of multiplicity is x^2. You would say is as x with a multiplicity of 2. For synthetic division, the first step is to do complete factorization, then list all the zeros. a+bi is a complex number, a-bi is a conjugate, fl them together. The rational zero test us all the factors of the constant (P) / factors of the leading coefficient (S). The first step for approximating zeros is to divide the interval [a,b] in half by finding its midpoint, m=a+b/2. Then, if f(a) and f(m) have opposite signs, then f has a zero in the interval [a,m]. If f(m) and f(b) have opposite signs, then f has a zero in the interval [m,b]. If f(m) = 0, then m is a zero of f. Finally, compute f(m). 

Piecewise Functions #5

A continuous piecewise function has no holes, no gaps, and is drawn without lifting your pencil. A discontinuous piecewise function is the exact opposite and has an end. The absolute value function has two pieces: below zero (-x) and from 0 onwards (x). For this type of piecewise function, once you graph the first side of it, you know the absolute value side, because it is a reflection from the first half of the graph. Given the title "piecewise," a graph can be in multiple pieces and not be connected at all! The greatest integer function (aka the floor function), has an infinite number of pieces and looks like steps or stairs when graphed. The pattern is continuous. 

Superheroes #4

I didn't really understand how to complete all of the missions, but I was able to do two of them. I think Lasy Straightedge was th best superhero. She moved smoothly along the X and Y graph and had great aim! Pawabawa was pretty cool and he also had great aim. The trickiest heros were the last four, because they were not the typical line or palabara. They made me frustrated and confused. 

Rational Functions #7

You can use the Rarional zero test in order to find zeros. First, take the final number without a variable and write out all the factors of it. This will be your numerator. Then, take all the factors of your leading coefficient and that will be your denominator. Once you have solved for all the possible zeros, you must plug each one back into the original equation. Yes, this will take a lot of time and you will want to scream as I did. When you find a zero that makes your equation zero, then you know you have found a rational zero! 

Zeros of a Function #6

For the Rational Zero Test, we first find all the possible zeros using P/S then test the zeros using the remainder theorem. A hint would be to graph to see all the possible points. The Rational Zero Tsst gives you the possibility of all the rational zeros. Once you find one, you can do synthetic division. Just like last week, we might have to use synthetic division multiple times. Since the work it really tedious, the Rational Zero Test is the best method to use. Remember both positive and negative parts of zeros!

Functions #3

Today, I learned that a function was an equation that you could put a in number called an input to end up with a solution which is the output. An example of a basic function equation would be f(x)=2x^2+1. A subsection of a function would be its domain. A domain is what you ask yourself, "what can I put in?" There are two ways to solve domains. First, if the problem has a square root you must set it to > or equal to 0, because a negative cannot be in a square root. Second, if there is denominator in the problem, set each denominator to zero, but keep in mind, the answer cannot be zero. If the answer is zero, the solution is undefined.

What I Learned in Week 1 #2

This week in Mathland, I learned how to graph intervals with brackets and parentheses. I also learned the properties of absolute value and how to find he distance between two numbers. The most recent lesson I learned was the Rectangular Coordinate System. I had to sketch (x,y) coordinates that satisfy two equations. The distance formula was also reviewed in order to solve for the different sides of a triangle. 

All About Me! #1

Let's see I grew up in San Gabriel and went to Temple City schools up until 7th grade when my parents moved me to Foothills is Arcadia. 

There are five people in my family, my mom, dad, older sister, and younger brother. I love being the middle child because I don't argue as much with them and am the most different in terms of personality. We get along, but I'm considered the "peacemaker."

I have played basketball since the first grade with my church then played in middle school and now high school. I also played soccer in kindergarten but quit when my best friend did for some reason. For about 4 years, I did ballet. 

My favorite colors are pink and purple and if I could, I would dress up every day of the week. People don't usually think of me as a girly girl, but I am! I love heals and fancy dresses and getting my hair done!

Oh! I loooove Disney. My favorite princess would be Belle because she loves to read and is kind to everyone. 

I am a very honest person and will tell you how it is. Some people don't like that, but it were up to me, I would rather have people tell me how it is rather than beating around the bush. I'm generally happy and get excited about random things. I love being with people and get freaked out when I'm alone.