Trig review week

This week we covered some of the basics of trig. A large part of trig is slaving and proving problems using the trig identities. There are multiple identities that are used, but the cool thing about them is that they can be moved around and formed to suit the user's needs. The point of proving something using trig is not of find the answer, because it it right in front of you. The point is to show how you get to the answer and what steps must be taken. This type of work is almost like a puzzle. There is not much arithmetic involved, so it feels like you are not even doing math. 

Parametric equations

Parametric equations are equations that express a bunch of different functions. When graphed, these equations can turn out to look like things ranging from a circle to a butterfly. Different types of parametric equations consist of circles, ellipses, hyperbole, parabola, spirals, butterfly curves, and many more. When graphed, these equations are not only forms of math, but art as well. There are infinite ways in which parametric graphs can look. 

Partial Fractions

Partial fractions are fractions in which the denominator consists of a variable adding or subtracting a number. The numerator of these fractions usually contain whole numbers. To add or subtract two partial fractions, you must find the common denominator, usually leaving you with quadratic equation on the bottom. You can Lao do this process in reverse. You would accomplish this by factoring the bottom of the fraction. 

Repeating decimals

Repeating decimals in essence are decimals that go on for ever. To convert a repeating decimals into a fraction, you would take the number after the decimal that is repeating, and put it over the number nine. For example, 0.7777 is 7/9. If, however, there are two numbers being repeated, the you would put them over 99. For example, you would write 45/99. If there is a number before the decimal, you would just add it to the fraction. 

Sequences and series

A sequence is essentially an ordered set of numbers. A very simple example of a sequence is 2, 4, 6, 8... A series too is an infinite set of numbers. Series can either be geometric or arithmetic. Geometric series are numbers that are multiplied by a certain number each time. Arithmetic series are similar, but they are just numbers that add or subtract a number each time. A series can have an endpoint and a starting point.

 

Parabolas

Parabolas are essentially graphed as "U"s. Any given point on a parabola is equally as far from the focus as it is the directrix. All parabolas are symmetric about their bisector or axis of symmetry. The equation of a parabola contains a y^2 and an x, or the other way around. The vertex of a parabola is the point at the very top of the bend. The can either be the highest, lowest, farthest left, or farthest right point of the graph, depending if it's positive or nevagtive and vertical or horizontal. Parabolas are used greatly in physics, such that they display the natural tendencies of gravity. 

Tower of Hanoi

When playing the tower of Hanoi puzzle, I noticed that to finish, one had to utilize all three pillars. I did not really find any specific rhythms or strategies for finding the fasted solution, but I did notice that one had to go against their instinct to complete it. Mathematical induction is seen in this puzzle through the reoccurring steps that include using all three towers. This proves that to get all the rings on the last tower, the middle tower must always be used. This is an inevitable  pattern that occurs whether there are 2 rings or 6.