ıllıllı ĹĂŴ ŐŦ ŚĨŃĔ ĂŃĎ ČŐŚĨŃĔ ıllıllı

Welcome back everyone~

Today we will be reviewing the laws of sine and cosine. Between the two laws, I would have to say the Law of Sine is quicker to set up and solve than that of the Law of Cosine.

To start off, the equation for the Law of Sine is:

The large (A) represents the degree of the angle, while the small (a) represents the length of the side. In order to find one of the sides, one would simply just "plug and chug".

On the other hand, the Law of Cosine is a bit more complicated. The equation for this law is shown below:

The cases in which one uses the law of cosine is when one is given three sides with no angles, or two sides and one angle. 

Both of these laws allow for the calculation of finding all of the angle and lengths of a certain triangle - usually ones that are not right angled.

I hope this was helpful.~

See you next time!~

|!¤*'~'*¤!| cнαρтεя 4 sυммαяү |!¤*'~'*¤!|

Ok Welcome to Mathland!

Today, we will be covering a summary on Chapter 4. Honestly, this chapter was quite long due to the numerous trigonometric equations we learned.

To start off, here are 14 basic trig identities:

These identities are the basis for the 12 other trig identities that encompass more difficult aspects. By using these identities, one is able to verify other identities.

The other 12 draw from these top 14^ and combine them to form angle identities.

Further below are examples each angle identity worked out.

Sum and Difference

 Double Angle Identity

 Half-Angle Identity

The bulk of Chapter 4 focused on such identities and in the following chapter 5, we will apply the Law of Sine and Cosine to these.

See you next time!~

]|I{•------» мr. υnιт cιrcle «------•}I|[

Hello and welcome again to Mathland!

This will only be a simple blog referencing to the unit circle.

This unit circle allows for one to easily locate the radians of a certain degree. 

In addition it allows for one to determine co-terminal angles as well as aid in solving trigonometric equations. Some facts about the circle are that the (x, y) coordinates correspond to (cos, sin). Also the full circle is 360 degrees. 

Moreover, the "All Students Take Calculus" acronym also apply within this.

That is all for today!

See you next time.~

ıllıllı ȚŖÏĠÖŅÖMËȚŖÏĊ ₣ŮŅĊȚÏÖŅŚ ıllıllı

Hello my dear guests of Mathland.~
I hope you are enjoying your stay so far, yes?

Today we will be conversing about Trigonometric Functions which encompass the use of triangles - usually the special triangles of 30-60-90 or 45-45-90. 

There are different names for finding the measure of each angle:

-Sine

-Cos

-Tan

or otherwise known as SOHCAHTOA.

There is also:

-Cotan

-Cosecant

-Secant

But those are only the inverses or reciprocals of the other.

However, there is an easier way of finding the measure or radians of each angle, and it is through the use of a unit circle (as shown below).

For example, if one was to ask for the cos of 45, the answer would be √2/2.

The various numbers of the angles are very diverse, similar to the bottles and deserts within the room Alice dropped in - oh you know, the drink me and eat me condiments?~

Oh well~

I do hope you enjoyed today. 

Good bye for now. Enjoy your day, ladies and gentlemen of Mathland.~

•°¯`•• verιғyιng ιdenтιтιeѕ ••´¯°•

Welcome back, dear guests on Mathland.~

Today we will be taking a tour to the singing flower garden.~ 

Along the way let me enlighten you with the intriguing steps in verifying identities.

You see, verifying identities, are not the easiest things to do and it requires time and practice to become adept at it.

There are a few rules that involve solving and verifying these trig identities:

1) You may only work on one side of the equation.

2) Choose the less complicated side to work with.

3) You may not cancel out any identities.

4)  Use the trig equations to simplify.

5) Using conjugates and factor by grouping will help significantly.

Here is just a simple and short example:

I hope this has helped you understand more about the verification of trig identities.~

And oh my, it looks like we have already arrived.  

Let us go and enjoy the singing flower garden.~

Good bye~

¯`•» ⓣⓐⓝⓖⓔⓝⓣ «•´¯

Welcome back to Mathland!

As promised, in this blog post I will be talking about the tangent function and graph.

Recalling from the past post about the other six different trigonometric identities, tangent is the one where is asks for the opposite side of an angle over the adjacent side.

The tangent graph however, is a unique graph since it involves asymptotes. Both the sine and cosine graph don't have asymptotes. 

Also the tangent graph stretches out to infinite.

The formula for this graph is y = a Tan (BX+C) + D

a = the amplitude

B = the period

D = the phase shift

To calculate the period, one sets  π/b. To find the asymptote, we use -π/2b and π/2b.

Now that is all for the tangent graphs. I hope this was helpful.

Thank you and see you next time!~

.o0×X×.ѕιne and coѕιne.×X×0o.

Welcome back to  Mathland!

Today we will be emphasizing on the aspects of sine and cosine. Do not be worried for we will be covering tangent in the next bog.~

Starting off there are six different trig identities and their corresponding sides:

Sine θ= opp/hyp

Cos θ= adj/hyp

Tan θ=opp/adj

Csc θ= hyp/opp

Sec θ= hyp/adj

Cot θ= adj/opp

In order to find any of these identities, drawing the triangle and labeling the corresponding sides will aid greatly.

Also, using the unit circle will greatly help in determining the answer when given the degrees. The unit circle is listed below for reference:

An important fact about the unit circle is that each quadrant also tells us what is positive and negative. 

Starting from Quadrant 1 and going counter clock-wise to Quadrant 4, the acronym "All Students Take Calculus" hint at the sign. All (Quadrant 1) indicates that all angles are positive. Students which stand for Sine indicate that in Quadrant 2, only Sine is positive. Similar in Quadrant Three - Take (Tan) - on tan is positive, etc.

That is all for the sine and cosine functions!~

I hope you enjoyed this small lesson.

See you next time.~

•]••´º´•» ¢нαρтєя 3 ѕυммαяу «•´º´••[•

[Admin: From here on, the blogs will just be lesson teaching.~ ]

In chapter 3, the following chapters encompassed polynomial functions, the division of polynomial functions, zeros and factors of polynomial functions, and then real zeros and factors of polynomial functions. In addition to those, we also emphasized on rational functions as well as approximating zeros. Within the chapter of polynomial functions, we learned how to divide them in two different ways. The first way was long division and the second was synthetic division.

Below are examples of each way of solving:

An important reminder of solving synthetic division is that the first numeric value is always brought down. After that step, then it is multiplied to the divisor and added to the other numbers. In addition, the remainder of either polynomial functions is always written over the divisor.

Thank you for your time.~

×X×. ŖÄȚÏÖŅÄĻ ₣ŮŅĊȚÏÖŅŚ .×X×

Oh my my~

It's been far too long, dear guests of  Mathland! 

Such a lovely time to see you again, for today we will be heading to the Hatter!

I must warn you though, since we are learning about rational functions, the Hatter might no quite be the most...."rational" about some things, yes?

As said, rational functions are usually derived from the equation of:

F(x) = P(x)/Q(x)

The "P" represents the factors of the ending coefficient while the "Q" represents the factors of the leading coefficient.

By using such an equation, the answer given could be similar to that of the vertical and horizontal asymptotes.

A much simpler way, to find the vertical asymptote, just set the denominator equal to 0.  The answer would be written as "x= (answer)", since the vertical asymptote is always along the x axis.

The horizontal asymptote however, you can find by comparing the leading exponents of the numerator and denominator.  If they are equal then we set the leading coefficients over each other, if the the top is greater than the bottom then DNE, and if the top is less than the bottom, y=0.

The slant asymptote on the other hand only occurs if the answer to the horizontal asymptote is DNE (does not exist) - or basically, the exponent of the numerator is greater than that of the denominator.

Finally, for holes, you factor the numerator and denominator to see if any of the parentheses cancel. If they do then you sent the canceled out parentheses to 0 and plug that answer into the left over un-factored equation.

Well, that will be all for today.~ 

I hope you enjoyed this little lesson as well as the visit to our dear Mad Hatter.

You never thought of him as a real person have you? 

It does seem quiet unrealistic for such a mad and i-rational man to exist...but then again...we are in the mad mad Mathland, yes?~

Very well then, I bid you all farewell and goodnight....~

•´¯`•» zeroeѕ oғ a ғυncтιon «•´¯`•

Oh, welcome back to Mathland, dear guests.

I hope you have not missed it too much!~

Today, we will be journeying to the White Queen's domain for a quick visit. On our way their, we will be discussing the zeros of a function.

Please do not get lost now.

Let us begin.~

Through the uses of synthetic and long division, we are able to derive the zeros of a function. First, you would plug the zero into the function given and if there is no remainder, than it means that the equation does indeed work. 

To find the zero of a factor - for example (x-3) - one must first set the equation equal to a zero and then either subtract or add determining on the sign it was previously.To clarify any confusion....

and

This indicates that once you plug in a zero to the function, the answer that come out will be the zero of the function.

However, when using long and and synthetic division, if a remainder appears once it is divided, then it indicates that the equation does not work. 

The significance of the zeros is that they are incorporated into almost every aspect of math.

Hmm...I believe that is enough for today.~

Well, it seems that we have arrived at her Highness's court.~

Oh my, it seems that the Queen is up to her mischievous cooking once again.

Good bye for now.~

º× pιece wιѕe ғυncтιonѕ ׺

Well well well... what a fancy place to meet.~

It's been quite some time, dear guests. Have you missed the lovely Mathland?

Would you like to get started?~ 

After all, this tea party is beginning to feel a bit dreary...

Shall we be on our way then?

Today, I will be taking you personally to the Queen's castle - but please be careful. She erupts into a rather large fit when things do not go her way. Please do not anger her or you just might possibly lose you head... 

Ah, it seems that we have arrived! I welcome you, dear guest, to the treacherous Heart Kingdom...

Today, we will be touring the palace and learning about the notorious "Piece Wise Functions". These functions will either be continuous or discontinuous when graphed. Meaning, when one places their figure on the line, if the finger must be lifted to transition, then it is discontinuous - and vice versa.  During the solving of these equations, there may be absolute value signs around the equations, to deal with these, you simply treat them as if they weren't there. After that, simply solve them regularly and out will come the answer. Next to the equations will be limits that tell you how far you may go: x less 0, equal to 0, greater than 0, etc. By using these limits, they specifically detail the placing of the graph. In addition -

"OFF WITH HER HEAD!!!!"

Oh my, forgive the interruption...it seems that the Red Queen is at it again. Well, this does certainly also help elaborate a point. The limits of the functions can be paralleled to how all the roses in the Queen's must be Red or else they will cease to exist because the Queen will behead those who did not paint them Red. This is partially why all the card soldiers are so diligent in making sure each and everyone is painted.~

I believe that is enough for today, since I seem to have to go pacify the Queen's tantrum currently. I do hope you enjoyed today. Please do visit again soon.~

Farewell lovelies!~

Have a wonderful rest of the night...

• ѕυperнero тranѕғorмaтιonѕ •

Welcome back lovely guests of wonderland.~

Today, we have a special event prepared by the Queen of Mathland!
We will be journeying with our new friends, the F(x) men! Allow me to introduce each of them:

And,

These superheroes are in charge of keeping Cartesanville safe from evildoers! By eliminating these baddies with their powers, they each demonstrate their own specific technique that is unique to them. Lady Straightedge has the function of f(x)= x, while Pawabawas the equation of f(x) = x^2. Robo-grow is f(x) =2^x, and Captain Abs is f(x) = |x|. Last but not least, Radical girl is f(x) = √x, and Bipolar Tommy is f(x) = x^2.  Below are some of their great accomplishments!~

All these superheros sure do remind me of how our dear Alice defeated the Jabberwocky and saved Mathland.~

Well, I must retire for now, I do hope you enjoyed this event prepared by our lovely Queen.

Farewell.~