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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.~

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[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.~

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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....~