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.