More Logarithms

Ok, so I havn't written a post in forever so I think its about time to share some things. We are now done our unit on logarithms, but I havn't said much about them on my blog, so here I go...

A natural logarithm(ln) is equal to loge. e= 2.71828... (it is an irrational number).

f(n)= 1+1/n g(n)= (1+1/n)^n

And thats all I have, wow. How sad. Maybe I'll google something later and add to this.

Logarithmic Laws

Logarithmic Laws include:

loga(MN)=logaM+logaN
loga(M/N)=logaM-logaN
logaM^x=x(logaM)

There is also the change of base theorem :
logbn=logan/logab

Work Period

Yesterday we had a substitute teacher, so we spent our class time working on any outstanding assignments and accelarated math. I found this time to be very useful because when I have alot of time in class to work on assignments I get more done, instead of being at home trying to figure things out all by myself.

Lately we have being doing logarithms and at first I was absolutely LOST! For so long I was so frustrated because nothing made sense to me, but then I went to tutoring on Tuesday and it just clicked. I like looking at problems and instantly knowing how to tackle it.

Logarithmic Functions

Logarithms are exponents.
A logarithmic function is the inverse of an exponential function.

Ex: Given y=2^x, the inverse is x=2^y, which is the same as y=log2x

Work Period

Today was early dismissal, so our math class was only 40 minutes long. Instead of learning something new, we were given time to work on our exercises that were previously assigned to us. There will be a homework check either tomorrow or Thursday on exercises 11-20, so make sure you get everything done! We should also be working on our Accelerated Math.

By the way, this week is Media Literacy Week!
Basically, its a week to promote the proper understanding and techniques of media education. You can learn more about it on this website: http://www.medialiteracyweek.ca

More Double Angle Identities

Today we were given a few more examples of double angle identities. I used to think that I understood what I was doing but now I'm not too sure. I get confused because I keep forgetting to look at my formula sheet and look at the identities and pick one that will work for the question I am working on. Boo :(

Finish Exercise 18 and work on accelerated math.

Work Period

Today we had a work period, I was very happy!
Test is moved to Thursday now, not Wednesday. Once again, I was happy :)

Double Angle Identities

Today we learned 5 new identities called Double Angle Identities. I understand this. Woo hoo...... ! Trig is Awesome.

Sum & Difference Identities

Today we learned 6 more identities called sum and difference identities. We used the example of 7pi/12 and how it can also be said as pi/3 and pi/4, so those two new values can be put into the corresponding sum/difference identity.

Homework: Exercise 16, questions 1-15 (except 3 & 5)

Extraneous Roots and More Identities

Well yesterday I was understanding almost everything that we were taught, but today I can kind of confused. I think all I have to do is somewhat memorize all of the identities and then it will help me with my work.

We were taught today that an extraneous root is a root that works on paper until it is checked and proved otherwise either through squaring or by other raised powers.

Trigonometric Identities

This is what I learned about identities:
-An identity is an equality that evaluates as true for any value of input.
-Trig equations that are not identities are called conditional equations
-Graphing is not adequate to prove "identity-ness", however, graphing is enough to disprove it

Basic Indentities are:
csc (x)= 1/sin(x)

sec (x)= 1/cos(x)

cot (x)= 1/tan(x)

tan (x)= sin(x)/cos(x)

cot (x)= cos(x)/sin(x)

Simplification Techniques to Assist Proving are:
1. Reduce complicated side to try to match the simpler side
2. Work each side independantly to some intermediate expression
3. Do addition/subtraction of the rational expressions
4. Do multiplication/division of the same rational expression
5. Simplify GCF's always
6. Factor! Factor! Factor!
7. Try multiplying both numerator/denominator by the same expression (to get known identities)
8. If possible, rewrite all trig functions as sin (x) or cos (x), look for patterns

Review

On Friday's class we were given Exercise 12 to work on. As we worked on our assignment, Mr Maks was in the cafeteria doing a homework check. We would each take turns going down one by one and we would chose 1 assignment at random to show him and he would give us a completion mark out of 10. The assignment I had to show him was Exercise 5, and that was an exercise that made noooo sense to me. I was given the mark 4/10, but Mr Maks gave me a website to check out and see if I could understand what I was supposed to do. I looked at the website and all of a sudden everything made sense! I finished the assignment and showed him again and recieved a better mark.

In today's class we got our tests back from 2 weeks ago, and I was happy with my mark. I got 46% and I actually thought that I did alot worse, so I was happy with the 46%. We were also given an answer key, so this way I can go back and correct mistakes, and also see how Mr Maks expects us to answer the questions. We were also given the answer key to the pre-test (with the superman cartoon on front) that we recieved last week which is similar to the test that we will be doing tomorrow! I'm not quite sure how I am going to do tomorrow, so I am deffinately going to look over the pre-test and then the answer key. Another thing I did today was finish the first 23 practice questions on Accelerated Math! I am testable in 2 objectives, so hopefully I will get that done soon and my mark will go higher. I am at 71% as of today, and I would like to keep it there or higher, no less than 70% for me!!!

Absolute Value Functions

Today we learned how absolute values can be understood as a piece-wise function (the function can be split into one or more pieces).

Definitions of an Absolute Value:
|x|= x if x is greater than or equal to 0
|x|= -x if x is less than 0

Example: When graphing y=|f(x)| :
|f(x)|=f(x) if f(x) is greater than or equal to 0
|f(x)|= -f(x) if f(x) is less than 0
The domain of |f(x)| is the same as the domain of y=f(x), but the range of y=|f(x)| will be f(x) is greater than or equal to 0.

I don't think I am making any sense... so I found this website that may help.
http://www.analyzemath.com/Graphing/Graph_Abs_Val_Func.html

Assignment: Exercise 11, Questions 1-12

Reflections and Graphing Reciprocals

Yesterday we learned about reflections (flips). I wrote everything down but I didn't quite understand what was going on. I will watch the recording tonight and hopefully understand what was taught to us yesterday.

Today we learned about reciprocal functions and how to graph them. We observed that:
1. f(x)'s x-intercept is 1/f(x)'s asymptote.
2. In the same interval (left to right, or right to left) f(x) is "greatering" and 1/f(x) is "lessering" (or vice versa).
3. It seems the at values of y=-1 and y=1, the graphs of f(x) and 1/f(x) are at the same place.

Translations

On Friday's class we started a new unit on Translations. We learned that translations are slides and only ever move left-right, up-down, or a combination of one of each. Today we continued learning about translations (horizontal stretches/compressions, and vertical stretches/compresssions).

Work Period

Yesterday we had a substitute teacher. Nothing new was taught to us, so we had a work period. I worked on my accelerated math and got most of it done.

Solving Using Real Numbers

Today we did some examples on solving trig equations over real numbers. I didn't really understand everything, but I do understand that the letter k represents an integer. We were also given a practice test to prepare us for our test next Thursday.

Work Period

Instead of learning something new today we were given the class to work on assignments that were assigned to us from previous classes. I really liked today's class because I usually don't have time to work on my assignments at school and I was able to get help from my teacher and other students in my class. I found today to be very productive because I finished my chart (and understand how I got all the answers) and I also went to math-tutoring today and that helped me with getting some assignments completed.

Random Math Review

Today we had a discussion about the chart that was due today and how that chart will benefit us on future tests. I havn't completed the chart, but I will get help tomorrow either during my spare or during math tutoring.
We also talked a bit about Accelerated Math, and Mr Maks went over some examples off of Jill's accelerated math sheet.

Today we were shown how to solve a trig equation by graphing it. I didn't understand about 75% if it, so I will have to wait until Monday to get the recording. I found that after I watch the recording of the class a second time I have a better understanding of what was taught. We also learned about a program on the computer called Graphmatica

More About the 6 Trig Ratios and What Not

Today we were given a better understanding of the 6 trig ratios and made a chart to show how to find the sine and cosine of angles without using a calculator. I remember in previous grades, people would say that in grade 12 you were not allowed to use a calculator, and I always wondered how that was possible because of the fact that you needed a calculator to show you how to find the tangent/sine/cosine of an angle, but now I see how that's possible! I didn't copy notes down in class but tomorrow I will get the notes off of the computer by saving them to my thumb drive and copy stuff out into my scribbler I keep specifically for notes.
Also today Mr. Maks went over how to write a proper (?) blog, so hopefully I am on the right track, considering this post is actually my experiences with math today, not just notes that I gathered.
Also Matt helped me out with fractions (like turning degrees to radians, and simplifying those fractions). I am absolutely terrible when it comes to fractions, so that's one thing that I will definitely have to practice.

Website

I came across this website while searching on Google and it has notes/examples on all the things we have been learning about the Unit Circle

http://www.analyzemath.com/Trigonometry.html#angles_trigonometry

Unit Circle Continued

6 Trig Ratios
Sin theta= y/1 ; Cosecant theta= 1/y

Cosine theta= x/1 ; Secant theta= 1/x

Tangent theta= y/x ; Cotangent theta= x/y

Unit Circle:

Radians & Degrees/ Unit Circle

Initial- starting point
Terminal- ending point

Complimentary- pi rads/2 (equals 90)
Supplementary- pi rads (equals 180)

Clockwise motion on a coordinate plain = negative
Counterclockwise " "= positive

Formula for arc length: theta= s/r
- theta= central angle in rads
- s= arc length
- r= radius

UNIT CIRCLE
Conforms to this x^2+ y^2= r^2 OR x^2+y^2= 1 (implies radius= 1 unit long)

Trigonometry

Radian- a unit of angular measure equal to the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle; equal to approximately 57.3 degrees (or 180/pi).

Degrees to Radian- ___degrees x pi/180
Radian to Degrees- ___pi x 180

Goals

- To have a minimum average of 70% in math
- To participate in school activities such as MOGA and Summer Activity Day
- To have a minimum average of 70% in each class for both semesters
- To not be afraid to ask questions
- To work at something until I understand it