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Senior Maths

Why use Radians and how do they work? Here is a good explanation from this website:

Why do we have to learn radians, when we already have perfectly good degrees?

Because degrees, technically speaking, are not actually numbers, and we can only do math with numbers. This is somewhat similar to the difference between decimals and percentages. Yes, “83%” has a clear meaning, but to do mathematical computations, you first must convert to the equivalent decimal form, 0.83. Something similar is going on here (which will make more sense as you progress further into calculus, etc).

The 360° for one revolution (“once around”) is messy enough. Why is the value for one revolution in radians the irrational value 2π? Because this value makes the mathematics work out just right! You know that the circumference C of a circle with radius r is given by C = 2πr. If r = 1, then C = 2π. For reasons you’ll learn later, mathematicians like to work with the “unit” circle, being the circle with r = 1. For the math to make sense, the “numerical” value corresponding to 360° needed to be defined as (that is, needed to be invented having the property of) “2π is the numerical value of ‘once around’.”

Therefore, I like to think of one circle turn as 2pi radians. That is, 2pi radians = 360 degrees. From this simple equation we can work out the size of any angle in radians or the how many degrees in a given number of radians.

Revision 2: Here is a useful video which helps explain the variations of a trig function (especially Period and Phase Shift). Click of the pic below to access the short YouTube video:

Another really useful explanation video can be found on YouTube here!

Revision 1: In class today we reviewed the three main trig functions; y=sinx, y=cosx and y=tan x. Then we looked at how changing the x coefficient altered the y=cosx graph (as this helps us understand a later question in the assignment). For example, the graphs below are of: y=sinx, y=sin2x and y=2sinx. Can you work out which one is which? The answer is given somewhere else on this post 😉

Graphing Trig Functions with Graphmatica –  note the gap between sin and x (y=sin x). Note: it is best to goto: Options–>Graph Paper and select Trigonometric for your graphing with trig functions. Graphmatica works in radian measure.

The Functions Project was handed out in class today and you will need to read through it carefully. There is also key information and resources to help you use Graphmatica for this assessment, including this one on YouTube. Please email any questions you have as you work through this project in class and at home.

In addition, sketch y=log x and y=10^x (Graphmatica makes this a breeze but you will have to type up as y=log(x)!) The answers for the three different y=sin functions in the May 14 post are: magenta is y=sin x; gray is y=sin2x (doubles up the frequency of the curve or, effectively, compresses it to repeat twice as often); red is y=sin0.5x (halves the frequency of sin x which means that it is lengthened); and cyan (blue) is y=2sinx (doubles the amplitude or height of the sin x curve). To summarise:

Magenta: y=sin x has an amplitude of 1 (goes up +1 and down -1 from the mid-line or x axis of the graph)
Gray: y=sin2x squishes the function by a factor of 2. So, the curve is being “cycled” twice as fast as y=sinx. This is shortening the period of the function.
Red: y=sin0.5x lengthens the cycling effect of the graph – that is, it cycles in double its length for y=sinx. This lengthens the period of the funtion
Cyan: y=2sinx has an amplitude that is 2x as tall. Hint: whatever the “sinx” is multiplied by gives you the amplitude of the function

Note: you can always do a simple table of values to plot the above 4 sine functions. More info on this excellent website.

April 27: As we reviewed the week’s work there were obvious gaps – e.g. not being able to identify functions from a set of ordered pairs (a set of x,y coordinates) and a few issues in accessing graphmatica, etc. These areas of weakness do need you to review the work done by referring back to the textbook and your notes. Some more practice examples of identifying functions can be found here;

The foundational work in Expanding and Factorising will be great preparation for the more visual work with Functions. Assessment includes a Project. Details soon, but here a good introduction to work through from mathwarehouse.com:

 

A Function is a unique kind of relationship. There are 4 types of algebraic relationships:

1) One-to-one
2) Many-to-one
3) One-to-many
4) Many-to-many

A Function is either Type 1) or 2) relationship. That is, for each element on the domain there is only one unique element mapped to on the range

7 Comments to

“Senior Maths”

  1. August 18th, 2012 at 8:44 pm      Reply tim Says:

    hey sir,
    any chance of getting a digital copy of the task sheet
    thanks


  2. August 14th, 2012 at 4:03 pm      Reply Drew Says:

    Hi sir what is the equation for normal distribution, the b=x positive minus. equation


  3. June 14th, 2012 at 6:59 pm      Reply Drew Says:

    Hi what questions did we have to do for homework


    • June 14th, 2012 at 11:50 pm      Reply h3maths Says:

      Hi Drew. I got to this comment quite late tonight. We are reading the early notes in Chapter 5 on “rates of change” and including another example. Thanks and see you tomorrow.


  4. June 1st, 2012 at 2:04 pm      Reply Kath Says:

    Hi Sir,
    Kath and Olivia here.
    Just writing to tell you that we do not understand the homework given, and will not be able to see you tomorrow as we have previous commitments.
    thanks, sorry for any inconvenience.


  5. May 30th, 2012 at 2:39 pm      Reply Stephanie Dunell Says:

    Cool blog sir!


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Largest number between o and 1 million which does not contain the ‘n’ is 88

 

Rotation SAT Problem: Answer: 4 (see: https://www.youtube.com/watch?v=FUHkTs-Ipfg)

 

Which number has its letters in alphabetical order? Answer: F O R T Y

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How long for the stadium to fill? 45 minutes.

Where are you? the North Pole

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Popular Math Problems Answers: 1, 1

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Worst Graph? Where to start. What a visual mess and even some of the lines merge and are impossible to follow. A graph is a visual display of data, with the goal to identify trends or patterns. This is a spider’s web of information which fails to show a clear pattern at all. Solution? Well, different colors would help, or why not group in two or three graphs where trends are similar?

Number of different nets to make a cube is eleven – see this link

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10 x 9 x 8 + (7 + 6) x 5 x 4 x (3 + 2) x 1 = 2020

NCEA Level 2 Algebra Problem. Using the information given, the shaded area = 9, that is:
y(y-8) = 9 –> y.y – 8y – 9 =0
–> (y-9)(y+1) = 0, therefore y = 9 (can’t have a distance of – 1 for the other solution for y)
Using the top and bottom of the rectangle,
x = (y-8)(y+2) = (9-8)(9+2) = 11
but, the left side = (x-4) = 11-4 = 7, but rhs = y+? = 9+?, which is greater than the value of the opp. side??
[I think that the left had side was a mistake and should have read (x+4)?]

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