WPP#13/14: Unit P-Concepts 6 and 7

Photo-credits:

http://24.media.tumblr.com/48a60113d127af3168cd9ff610d86684/tumblr_my0biwlZRg1qmxwggo1_r2_500.jpg

Hazel grace is located 135 miles due south of Augustus Waters. Destiny is calling out for their first meet in order for them to meet and fall in love. Destiny's call indicates that Hazel Grace is located at a bearing of 60 degrees; Destiny's call indicates that Augustus Waters is located at a bearing of 131 degrees. How far are Hazel Grace and Augustus from Destiny?

Photo-credit: lovely, hand-drawn pictures by Mrs. Ventura

Two planes leave the airport at the same time, traveling on courses that have an angle of 125 degrees between them. If the first plane travels at 35 miles per hour and the second travels 28 miles per hour, how apart are the two planes after traveling for 5 hours?

Photo-credit: lovely, hand-drawn pictures by Mrs. Ventura

BQ#1- Concepts 1 and 4

1. Law of sines-Why do we need it? How is it derived from what we already know?

Photo-credits: lovely, hand-drawn pictures by Mrs. Ventura

Using the picture from above, we will go step-by-step of the derivation of the equation. First things first, we must know that the law of sines is needed to solve missing parts of a non-right triangle. A regular right triangle can be easily solved because it has all the needed components, that being, most importantly, a right triangle. So, splitting a non-right triangle we get two right triangles. We solve for both angles A and C using sin. From there we get sinA=h/c and sinC=h/a. We take it one step further and we multiply by c  and a ,for the two separate equations, in order to get h alone. Having h alone for both equations, we then combine them and move on over the a and c to their appropriate sides. From there we get the law of sines. Of course the law of sines is not just restricted to a or c, it also works for b as well. The law of sines fluctuates according to what specific information you are looking for.

Photo-credits: lovely, hand-drawn pictures by Mrs. Ventura

4. Area formulas- How is the "area of an oblique" triangle derived? How does it relate to the area formula that you are familiar with?

The area of an oblique triangle is derived from the area formula and a combination of a trig function. The area formula is A= 1/2bh. With a non-right triangle, trying to solve the area is much more difficult. Therefore, we split the triangle with a perpendicular line and get two triangles. From there, we find angle C. SinC=h/a, we multiply a to the other side in order to get h alone. Moreover, we plug in h into the original area formula. That, is the equation for solving the area of an oblique triangle.

It's similar to the area formula we know, except we manipulate it, in order to get what we need. The only difference is since we do not have h we use our sources and use it to the best of our abilities.

Below are other equations that are used to solve oblique triangles:

Photo-credits: lovely, hand-drawn pictures by Mrs. Ventura

BQ #7- Where does the difference quotient come from?

Photo-credits: lovely, hand-drawn pictures by Victoria

Looking above at the image, we notice that our first points the x-value is x and our y-value is f(x). Technically looking at the graph not a certain distance is given so that's the reason why we use f(x). For the very same reason the next set of points x-value is x+h and the y-value is f(x+h). The sole purpose of this graph is to find the slope tangent line on the two specific points. To find the slope of the tangent line we must use the slope formula which is m=(y2-y1)/(x2-x1). We plug in the points to the formula and continue on simplifying as much as we can: cancelling/combining like terms, simplifying, etc. Once we have finished the whole lovely process we get the difference quotient! Voila! The difference quotient is extremely helpful in finding derivatives for a certain graph. A visual is displayed below for the process of plugging in to the points to the slope formula ---> to becoming the difference quotient.

Photo-credits: lovely, hand-drawn pictures by Victoria

BQ#6: Unit U-CALCULUS INTRO WOOO

1. What is continuity? What is discontinuity?

 

Photocredits: http://www.mathsisfun.com/calculus/continuity.html

In order for a graph to be continuous, first of all, must be predictable. Secondly must not have any jumps, holes or vertical asymptotes. Lastly, must be drawn without lifting the pencil off the paper. All of these elements can be seen in the pictures above. Now discontinuities are the exact opposite of this. But do note that there are two different types of discontinuities: removable and non-removable. A question frequently asked is why are there two groups and that my friends is because removable are when limits do exist and non-removable discontinuities do not exist. One removable discontinuity is point and the limit here does exist because although it does not reach the intended height it does reach a certain height the actual height and that is seen through the value. The three non-removable discontinuities are jump, oscillating and infinite. A limit cannot exist in jump because there are differences in left and right graphs and they do not meet. A limit does not exist at oscillating because, well, it cannot be found. The graph is too out of control to actually find where the limit exists. In an infinite discontinuity a limit does not exist due to unbounded behavior.

Photocredits: Unit U SSS Packet by Mrs. Kirch

 2. What is a limit? When does a limit exist? When does a limit not exist? What is the difference between a limit and a value?

A limit is the intended height of a function. A limit exists when a graph is continuous or as previously stated when, no jumps, holes or vertical asymptotes are present. A limit does not exist when there are left and right differences (jump), when the graph is wiggly (oscillating) and when there is unbounded behavior (infinite). All of these stated are elements of non-removable discontinuities. Lastly, there is a BIG difference between a limit and value. A limit is the intended height of a function, BUT the value is the actual height of the function. When solving equations it is extremely important to know the difference between the two knowing it will affect your answer (ya know, whether you get it right or wrong. OBVI you wanna get it right soo...know the difference!).

3. How do we evaluate limits numerically, graphically, and algebraically?

Numerically- To solve limits numerically a table would be necessary and key to help you solve. The limit statement that is said verbally. Is "the limit as x approaches # of (blank) is #. Or as seen in the image below

Photocredits: Unit U SSS Packet by Mrs. Kirch

Graphically-

Photocredits: Unit U SSS Packet by Mrs. Kirch

As seen above a limit was solved graphically. It is vital to know the difference of the value and limit here because solving it graphically makes it more difficult. I through M are all limits for I and M since they do not exist a b/c is necessary in order to fully understand why the limit DNE. Sometimes, because a limit does not exist, it does not mean that a value is present. So when solving a limit graphically it is HIGHLY important to pay attention to all the little details, they are very important.

Algebraically- There are three ways to solve a limit algebraically those being: direct substitution method, dividing out/factoring method, and rationalizing/conjugate method.

FIRST method to try out would be the substitution method in which you substitute all the variables with the give number that x approaches in the limit statement. If it turns out to be undefined the next step to trying to solve it out is using the dividing out/factoring method. In this method we factor out any long equations and then find common factors and divide them out. With what is left from there we substitute the given number x approaches in the limit statement given and if we get an answer AWESOME-SAUCE and if we don't we go for our last and final method that being: the rationalizing/conjugate method! In this method you multiply by the conjugate in the denominator to eliminate some things and make life much easier, this method is the ONE. And most of the time works out fine so whenever the other two methods do not make your life easier, come to this one! If more help is needed in understanding the last method, look at the image below:

Photocredits: Unit U SSS Packet by Mrs. Kirch

BQ#4: Unit T Concepts 1-3-Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?

Photocredits: desmos.com

Yes, tangent and cotangent are inverses of each other but, that does not fully explain why exactly a tangent graph is uphill and a cotangent graph is downhill. Let's start off with tangent first and why it's uphill. Tan=sin/cos in order for tan to be undefined and have asymptotes cos must be zero in certain points of the unit circle: pi/2 (90degrees) and 3pi/2 (270degrees). Co-tan, unlike tan, has asymptotes in other points on the unit circle, it's asymptotes are on: 0pi, pi (180 degrees), 2pi (360 degrees). And that completely answers our question. Tangent graphs and cotangent graphs go in two different directions because of the different asymptotes, they are undefined in different areas, so, therefore, they go in different directions.

Photocredits: desmos.com

BQ#3:Unit T-How do graphs of sine and cosine relate to each of the others?

a. Tangent?

                                                         Photocredits: desmos.com

In order to fully understand how all trig functions seen on this graph relate, lets go quadrant by quadrant. In the first quadrant, sin, cos and tan are positive. So, therefore, all of the graphs at this point are above the x-axis showing they are positive. In the second quadrant, it's quite different sin is the only trig function that's positive in this quadrant the other two are negative. In quadrant three, tan is the only positive trig and the other two are negative. Lastly, in quadrant four, cos is the only positive and the other two are negative.

So, what's the big deal with these asymptotes and why are they placed in the places they are placed? Tangent has asymptotes because at certain points such as pi/2(90 degrees) and 3pi/2 (270 degrees) tan's ratio (x/y) is undefined. That is why the tangent graph looks split in between sections, the tangent periods don't touch the asymptotes but do get very close. 

b. Co-tangent?

                                                                                   Photocredits: desmos.com

Cotangent's graph is very similar to tangent, positive at all the same parts and negative at all the same parts, except that they go in two different directions. Why? Well, because yes they're inverses of each other but they have asymptotes at different points on the unit circle. In the Unit Circle tan's ratio is undefined at 0, pi (180 degrees) and 2pi (360 degrees). Therefore, the co-tangent graph goes downhill.

c. Secant?

                                                                                   Photocredits: desmos.com

Secant, the beautiful secant. So, what's up with this so-called secant? In the first quadrant, all three, sin, cos and secant are positive. In the second quadrant, sin is the only one positive, the other two are negative. Furthermore, the third quadrant, all (sin, cosine and secant) are negative in this quadrant. Lastly, in the fourth quadrant, both cos and secant are positive, and the only one negative is sin.

The asymptotes for secant are on pi/2(90 degrees) and 3pi/2 (270 degrees). Those are the specific asymptotes for secant because that is where secant's ratio is undefined.

What seems so odd is that the asymptote of secant is connected with cos, why is that? That is because in simple words secant is the inverse of cosine! 

d. co-secant?

                                 Photocredits: desmos.com

Co-secant is the inverse of sine. In the first quadrant, all three, sin, cosine and co-secant are positive. In the second quadrant, sin and co-secant are positive and cos is negative. Moreover, in the third quadrant, all the three trig functions are negative. And lastly, in the fourth quadrant cos is the only one positive, the other two (sin, co-secant) are negative.

Much like co-tangent, co-secant has certain asymptotes because that is where its ratio is undefined. Co-secant is undefined at at 0, pi (180 degrees) and 2pi (360 degrees).

And just like secant, co-secant is attached to sin because its the inverse of co-secant!

Thank you for reading my blog. Comments, concerns, questions? Feel free to call me *wink*. Just kidding, comment below..

BQ#5: Unit T Concepts1-3-Why do sine and cosine NOT have asymptotes, but the other four trig graphs do?

First question that should be answered is: how can you get an asymptote? When a trig function is undefined is where an asymptote would appear on a graph. (Okay, got that cleared.) We must know that in the Unit Circle r=1. So therefore, sine and cosine can never be undefined because they would always have a one underneath in their ratio. Unlike sine and cosine, co-secant, secant, tangent and cotangent CAN have asymptotes. Why? Well for one, they do not have "r" underneath as their denominator in their ratio. Meaning, they can possibly have one of their values on the bottom as 0. For instance with secant, its ratio is r/x, we know that "r" is 1 and possibly if "x" is 0, our answer would be undefined. For co-secant, secant, tangent, and cotangent there are many possibilities that an answer can be undefined and that is why these trig functions have asymptotes and, sine and cosine do not. Sine and cosine will never be undefined nor have an asymptote. 

ANY QUESTIONS? FEEL FREE TO COMMENT BELOW, THANK YOU.