BQ #7: Unit V: Derivatives and the Area Problem

1.  Explain in detail where the formula for the difference quotient comes from now that you know! Include all appropriate terminology (secant line, tangent line, h/delta x, etc).

    The difference quotient is very significant in calculus and it is known as the derivative. The derivative is the slope of all tangent lines on the graph. The difference quotient is very helpful in finding all the possible slopes and curves that appear on any type of graph. A tangent line touches the graph once, whereas a secant line touches the graph twice. The y axis is notated as f(x), while the x axis is simply notated as x and x+h. Keep in mind that the letter h can also be written as delta x. The derivative, moreover, is written as f'(x) or "f prime of x". In this unit, you are not only doing the difference quotient, but you are also using it to determine the limit as h approaches 0. Once you find your derivative, you can use it to find specific values, slope, or the y=mx+b equation. Remember, the difference quotient is f(x+h) - f(x) divided by the letter h. This is similar to Y2-Y1/ X2-X1 (slope formula).

Example of a graph and plotted points using f(x)

Derivative

Tangent & Secant Line

Reference

http://clas.sa.ucsb.edu/staff/lee/Secant%20and%20Tangent%20lines.gif

http://clas.sa.ucsb.edu/staff/lee/Tangent%20and%20Derivative.gif

http://www.teacherschoice.com.au/images/derivative_secant_1.gif

   

BQ #6: Unit U

1. What is continuity? What is discontinuity?

    A continuous function is at most predictable. It consists of no jumps, no breaks, and no holes. You can draw a continuous function without lifting up your pencil. A discontinuity has two major families: removable discontinuities and non-removable discontinuities. The removable discontinuities only consists of a point discontinuity where a hole exists. The non-removable discontinuities are jump discontinuity (different left/right), oscillating behavior (wiggly), and infinite discontinuity (unbounded behavior and vertical asymptotes).

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 the function. A limit exists at removable discontinuities, which also refers to point discontinuity or hole. Because from left and right, it is the intended height for both sides and therefore, the limit does exist. A limit does NOT exist at non-removable discontinuities, which includes jump discontinuity, oscillating behavior, and infinite discontinuity. A limit does not exist at jump discontinuity because of different left and right. The limit does not exist at oscillating behavior because it is wiggly and has no single value since it does not approach it. The limit does not exist at infinite discontinuity because of unbounded behavior due to the vertical asymptote.

     While the limit is the intended height of the function, the value is the actual height of the function. In the picture below in the first graph, the limit exists but the value does not. It is undefined since f(c) is a hole. In most cases, the value is undefined when there is a hole. In the second graph, the limit exists but the value exists somewhere else. Although the intended height is known, the actual height has already been defined. In the third graph, the limit does not exist because of jump discontinuity. The value, though, still exists. It exists at only one of one side limits or at the closed circle.

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

Algebraically

    To solve limits algebraically, there are three different methods: direct substitution method, factoring method, and conjugate method. In direct substitution method, you simply plug in the number given and see what you get. In the factoring method, you have to factor out both the numerator and denominator. Then, cancel common terms in order to remove the zero in the denominator. In the conjugate method, you simply rationalize the either the numerator or denominator depending where the radical is. When using these methods, always try direct substitution method first! If you plug in x and get 0/0 (indeterminate form), then you would either use the factoring or conjugate method.

Numerically

    To find limits numerically, you will need to make a table. Before finding the limit, start from left and right on both ends of the table. Add/subtract a tenth of what is given. Then, write down the numbers that will eventually get smaller and smaller. For example, 2.9, 2.99, and 2.999. 

Graphically

   To determine the limit graphically, place your fingers on the graph to the LEFT and RIGHT of where you want your limit to be evaluated. If your fingers do not meet, then the limit does not exist due to different left/right, wiggly, or unbounded behavior because of vertical asymptotes.

REFERENCE:

Mrs. Kirch's SSS packet

BQ #4: Unit T Concept 3

Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?

   The "normal" tangent graph is uphill because of its asymptotes, which is located at pi/2 and 3pi/2 and so forth. The quadrants is positive, negative, positive, and negative. Tangent is uphill based on its asymptotes. Meanwhile, cotangent is downhill because its asymptotes is located at 0 and pi and so forth. In a way, it's asymptotes is "shifted" away from the "normal" tangent graph's asymptotes. Although the quadrants still apply the same for cotangent, it is a downhill because of where the asymptotes is located. Remember, tangent and cotangent have different asymptotes because of its ratio and the denominator in the ratio (sine or cosine will equal to 0).

BQ #3: Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others?

1. Tangent

   Tangent has a ratio of (sin/cos). When the tangent graph is labeled based on the quadrants of the Unit Circle,  the positive and negative sign of tangent is determined by the positive and negative sign of sine and cosine. If both sine and cosine is positive on the same quadrant, then tangent will be positive, ending up with an uphill on the graph. If sine is positive and cosine is negative on the same quadrant, then tangent will be negative, ending up with the downhill on the graph. If cosine is negative and sine is positive, then tangent will be negative, which will still have a downhill on the specific quadrant of the graph. If both sine and cosine is negative, then tangent will be positive, which will have an uphill on that specific quadrant of the graph. Tangent has asymptotes when cosine equals to 0. Cosine equals to 0 at pi/2 and 3pi/2, which is where the asymptotes are located and goes on continously at the graph.

2. Cotangent

   Cotangent has a ratio of (cos/sin). Similarly, the cotangent graph is based on positive and negative sign of cosine and sine. However, the difference between tangent and cotangent is that it will have a downhill shape in the beginning of its period rather than an uphill shape like tangent. The reason for its downhill shape is because of where the asymptotes are located. Cotangent has asymptotes when sine equals to 0. Sine equals to 0 at 0 and pi on the Unit Circle. Thus, this is where the asymptotes will be located for cotangent.

3. Secant

   Secant is the reciprocal of cosine and has a ratio of (1/cos). Because cosine is the denominator and can equal to 0, then secant will have asymptotes. Secant has asymptotes at pi/2 and 3pi/2 and so forth. If secant is positive, then the graph will go uphill. If secant is negative, then the graph will go downhill. In other words, if cosine is positive or negative on the graph, so will the secant.

4. Cosecant

   Cosecant is the reciprocal of sine and has a ratio of (1/sin). Because sine is the denominator and can equal to 0, then cosecant will have asymptotes. Cosecant has asymptotes at 0 and pi and so forth. If sine is positive, so will the shape of cosecant of the graph. In other words, if cosecant is positive or negative on the graph, sine will also have to be postive or negative respectively.

BQ #5: Unit T Concepts 1-3

Why do sines and cosine NOT have asymptotes, but the other four trig graphs do?

    Sine and cosine have a ratio of (y/r) and (x/r) respectively. The variable r equals to 1 on the Unit Circle, which is the reason why sine and cosine will never reach undefined since its denominator is 1. Cosecant and secant have a ratio of (r/y) and (r/x) respectively. Moreover, their ratios are reciprocals to sine and cosine. It is possible for cosecant and secant to have an undefined answer since the denominator does not equal to one. Tangent and cotangent have a ratio of (y/x) and (x/y) respectively. None of these two trigs have the variable "r" in their ratio. Thus, there is a likely chance that they will have an undefined answer. Remember, undefined equals asymptotes. All four trigs, except for sine and cosine, can divide by zero. This causes it to be undefined and have asymptotes on the graph.

Reference:

Mrs. Kirch's SSS Packet

BQ #2: Unit T Concept Intro

1. How do the trig graphs relate to the Unit Circle?

   The trig graphs relate to the Unit Circle because it uses the positive and negative values of the quadrants in order to define how a period for a certain trig may look like. Each trig has its own unique shape. Nevertheless, in a trig graph, the Unit Circle basically unfolds into a line. The points and  the radians from the Unit Circle is still very similar in a trig graph when reading from left to right. A sine trig graph shows an uphill from the two first quadrants on Unit Circle and then downhill for the remaining quadrants. Cosine shows an uphill, downhill, and then uphill again because of the positive, negative, negative, and positive value on the quadrants. Tangent/cotangent both have an uphill and downhill. The trig graph for tangent/cotangent have a shorter period because the positive and negative on the first two quadrants is similarly repeated again in the third and fourth quadrant.

2. Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

   On a Unit Circle, sine begins with a positive, positive, negative, and negative value on the quadrants accordingly. Cosine begins with a positive, negative, negative, and positive value on the quadrants accordingly. The period for sine and cosine is 2pi because it takes four quadrants to repeat the same pattern over and over again. Meanwhile, tangent/cotangent has a period of pi because it already has a positive and a negative value on the first two quadrants on the Unit Circle. Thus, it only takes pi to repeat the pattern over and over again.

3. Amplitude? - How does the fact that sine and cosine have amplitudes of one (and the other trig functions don't have amplitudes) relate to what we know about the Unit Circle?

   Sine and cosine have amplitudes of one because in the Unit Circle, sine and cosine cannot be greater than 1 or less than 1. Otherwise, it will be considered as "error" or "undefined". Since sine and cosine has a ratio of y/r and x/r respectively (and r = 1), sine and cosine can only fit under the rule of greater or less than 1 and
-1.

Reference
Mrs. Kirch's SSS Packet

Reflection #1: Unit Q: Verifying Trig Identities

1. What does it  mean to verify a trig function?

    To verify a trig function means to be able to get a specific answer according to what's given within the problem. Verifying a trig function takes quite an amount of time since your goal is to try different steps and plug in any Ratios, Reciprocals, or Pythagorean Identities into the problem. Remember, verifying means that your answer or final result needs to match to what is given to you. Verifying can go along with the same steps as simplifying.

2. What tips and tricks have you found helpful?

   The most helpful tip there is to try to change your trigs in terms of sine and cosine. Another helpful tip is to memorize all of the Ratios, Reciprocals, and Pythagorean Identities. At times, it is helpful if you separate the monomial fractions or combine the binomial fractions. When in doubt, you can always take the greatest common factor or least common denominator so in the end, your denominator can cancel.

3. Thought process and steps when verifying a trig function

   Personally, when I verify a trig function, I tend to change the trigs into sine and cosine. If this step does not work and my denominator is a binomial, then I would do the conjugate to the denominator and numerator. My other option when verifying a trig function is to separate the fractions (if any) and change it in terms of sine and cosine. It is also very useful to take the greatest common factor because there might be a chance when that GCF can cancel.

SP #7: Unit Q Concept 2: Finding All Trig Functions (using identities and SOH CAH TOA)

   In this problem, it is given that tan(x)= 2/3 and sec(x) < 0. It is also given to us we are going to use Quadrant III and the values to start our problem. Before we start solving, it is better to decide first if the sign is going to be negative or positive based on what is given to us. It can only work in ONE quadrant and only ONE of these quadrants can work (in this case it is tangent or Quadrant III).

  Reading from left to right, this is one way to find the missing pieces accordingly. When solving for cotangent, you can use the reciprocal identity. Now that you know what cotangent is, you can use the Pythagorean Identity. Plug in the value of cotangent, add, and take the square root. From there, you will find your value for cosecant. Because you know the value for cosecant, you can use it to find the value for sine. Thus, you can use the reciprocal identity. Remember, you must rationalize in order to get rid of the square root on the denominator. Now, if you look back to see what trig you are missing, you find that you still need to solve for secant and cosine. Since tangent was given to us, we can use the Pythagorean identity to find secant. Once we find the value for secant, we can use the reciprocal identity to find cosine.

  We can use our knowledge of SOH CAH TOA to find/check our values. This picture demonstrates the ratio and steps to get the value. Remember, "r" does NOT equal to 1. If you look back at the problem, it is given to us that "r", the radius, is 2 radical 13. 

I/D #3: Unit Q Concept 1: Using Fundamental Identities to Simplify or Verify Expressions

INQUIRY SUMMARY ACTIVITY

1. Where does sin^2x + cos^2x=1 come from to begin with?

     To start off, an "identity" is proven facts and formulas that are ALWAYS true. The Pythagorean Theorem is an identity because it has been proven that in a triangle, for instance, a^2 + b^2= c^2. For this unit, the Pythagorean Theorem can be consist of x,y, and r similar to that of a, b, and c according to the Unit Circle in quadrant I.

   Because a^2 + b^2=1, we divide r^2 on both sides. We can use our knowledge from the Unit Circle to determine what is (x/r)^2 and (y/r)^2. From there, we have found our first Pythagorean Identities. 

  Cos^2x + sin^2x=1 is referred to as a Pythagorean Identity because we used the Pythagorean Theorem to find it. Basically, we substituted variables and connect that to what we have learned. 

   To be sure if this Pythagorean Identity is true, we can use the angles (30, 45, or 60 degrees) from the Unit Circle and substitute those values. This picture shows an example of using the 60 degrees. 

2. How to derive the two remaining Pythagorean Identities from sin^2x + cos^2x=1

   The picture above demonstrates the various steps of how to get tan^2x + 1 = sec^2x. First, you will have to divide both sides by cos^2x. Then, with your memory of the ratio identities, reciprocal identities or Pythagorean Identies, you would substitute it into the equation. The identity with Secant and Tagent is tan^2x + 1 = sec^2x.

   The picture above shows the various steps of how to get 1 + cot^2x = csc^2x. You divide sin^2x on both sides. Then, using your memory, plug in what you know. At last, you should be able to find the identity for Cosecant and Cotangent. 

INQUIRY ACTIVITY REFLECTION

1.THE CONNECTIONS I SEE WITH UNIT N, O, P AND Q SO FAR ARE ratios that we use again from the Unit Circle and how we use quadrant I to know what variables to plug into the equation that we are trying to find.

2. IF I HAD TO DESCRIBE TRIGONOMETRY IN THREE WORDS, THEY WOULD BE difficult but fun.

WPP #13 and 14: Unit P Concept 6 and 7: Law of Sines and Law of Cosines

This WPP #13 and 14 was made in collaboration with Ashley V. and Sarahi L. Please visit the awesome blog post here!

BQ #1: Unit P Concept 3 and 4: Law of Sines and Area of an Oblique Triangle

Law of Sines:

1. Why do we need it? How is it derived from what we already know?

    The Law of Sines is used for non-right triangles. We cannot use the Pythagorean Theorem for non-right triangles. The Law of Sines can be used when it comes to non-right triangles with AAS and ASA. Remember, just like sine on the Unit Circle, the Law of Sines cannot be greater than 1 or less than 1. To make sure where the Law of Sines is derived from, draw a non right triangle.

   To make a right triangle, draw a perpendicular line from the top of the triangle to the bottom and name that "h." Now that you have two triangles, you can use SOH CAH TOA. When doing using the soh cah toa, you found that sinA= h/c and sinC= h/a. To get rid of "h", you multiply both sides by its denominator. The final answer is cSinA= aSinC. To get the Sine angle alone, divide both sides by "a" and "c".

Area of an Oblique Triangle:

1. How is the "area of an oblique" triangle derived?

   The area of an oblique triangle is derived when you substitute sinC=h/a, which is h=asinC, into the regular area equation of a triangle, A=(1/2)BH. By doing that, you will get 1/2b(asinC). Remember, we are still using the trig equations or SOH CAH TOA to get "h", the perpendicular line.

How does it relate to the area formula you are familiar with?

   This relates to the area formula that we are familiar with because you are simply using sin, cos, or tan to find "h" which is the perpendicular line that we "dropped" since these triangles are not right triangles. The area of an oblique triangle needs to contain two sides and an included angle. Therefore, it has to be SAS in order to use the area of an oblique triangle.

References:
Mrs. Kirch's SSS packet

WPP #12: Unit O Concept 10: Solving Angle of elevation and depression word problems

Problem: It is that time of the year again... for Superbowl Puppies! The husky, on its paw ready to compete the game, is about to jump on a stool. The angle of elevation to the top of the stool is 33 degrees and 8 minutes. a) If the base of the stool is 12 feet from the puppy, what is the height of the stool (to the nearest foot)? In the next round, the same puppy has to slide down the mini puppy slide from where he is standing (the stool). The puppy is standing on top of the slide looking down at the ground. The vertical distance from the ground to the puppy's eyes are 6 more feet than the height of the stool. The angle of depression is 54 degrees and 16 minutes. b) How long is the slide that the puppy is going to slide on(to the nearest foot)?


                                     

http://www.visualphotos.com/photo/1x8325432/siberian_husky_puppies_in_bucket_AFA-Z-01502.jpg

I/D #2: Unit O Concept 7-8: How can we derive the patterns for special right triangles?

  The activity of how to derive the patterns for special right triangles includes a square and an equilateral triangle. Based on the angles of these shapes, you can easily define what type of special right triangle it is, whether it is a 45,45,90 triangle or a 30, 60, 90 triangle. Each side length will be equal to one. We will used what we have learned about special right triangles and relate it to different formulas (Pythagorean Theorem) or concepts (Unit Circle) that we know. 

INQUIRY ACTIVITY SUMMARY

1. 30-60-90 triangle

  This is an equilateral triangle that has a sum of 180 degrees. Since all angles and sides are equal, each angle has to be 60 degrees. To get a special right triangle out of an equilateral triangle, you will need to split the triangle in half vertically. Side A will have an angle of 30 degrees, side B with 60 degrees, and side C with 90 degrees. According to the 30-60-90 special triangle, each side is equal to n, n radical 3 and 2n accordingly. Remember, "n" is a variable. For instance, if a triangle has a side length of one more than one, the ratio for the triangles will be the same. The variable "n" tells us the the approximate length of the triangle.

  The side length of the triangle is one. Equilateral triangles all have the same length and angles. Thus, each side for this triangle is one. Since the special right triangle only occurs when the equilateral triangle is in half, then the side A will have to be equal to 1/2. As a fact, 90 degrees equals to 2n. 1/2 (which is n) multiply by 2 will give you 1 as your hypotheses length. To find side B, you can simply use the Pythagorean Theorem. Side B will then eventually equal to radical 3 over 2. Another way to solve for side B is to plug 1/2 (which is n) to n radical 3. This step will also give you the same side length for B.

2. 45-45-90 triangle

  It is given to us that the square has a side length of one. A square has all equal sides and angles. Since a square has a sum of 360 degrees, each angle is 90 degrees. To get a special right triangle out of a square, you must cut the square in half diagonally. The special right triangle will be 45-45-90. Side A and B are equal to one (a given). In a 45-45-90 triangle, it is n, n, and n radical 2 accordingly. Since one is your "n", you can simply plug in 1 to n radical 2, which will give you radical 2 for the hypotheses. 

  You can also use the Pythagorean Theorem to find side C. Side A and B are both equal to one. As you solve for c^2, you get radical 2. Radical 2 cannot be simplified any further, so radical 2 is the side length for your hypotenuse. Remember, "n" is only a variable. You could have also plug in your side A or "n" to find the length of the hypotenuse. The variable "n" is a ratio for the special right triangle whether the side length is one or more than one. Thus, it is like a constant. The variable gives the special right triangle an approximate length. 

INQUIRY ACTIVITY REFLECTION

1. SOMETHING I NEVER NOTICED BEFORE ABOUT SPECIAL RIGHT TRIANGLES IS how we can derive these triangles to fully understand why the variable "n" is so important. Once you know how to apply "n" and the Pythagorean Theorem, it becomes a lot easier to find the side length.

2. BEING ABLE TO DERIVE THESE PATTERNS MYSELF AIDS IN MY LEARNING BECAUSE I can fully understand how a special right triangle functions and use this knowledge to apply to concept 7 and 8. I can also fully recognize where these special right triangles come from. I will forever remember the variable "n" and the patterns that these triangles contributed.

I/D #1: Unit N Concept 7: Knowing the degrees, radians, and ordered pairs of a Unit Circle

   The special right triangle (SRT) and the unit circle relate to each other in a way that the special right triangle of 30, 60, 90 and 45, 45, 90 can be found in the first quadrant of the unit circle. Moreover, it reveals the ordered pairs on the other three quadrants, depending on its sign (positive or negative). Special right triangles have a hypotenuse of 1 and the unit circle has a radius of 1. Thus, special right triangles can be located in all of the quadrants on the unit circle.

INQUIRY ACTIVITY SUMMARY

 1. 30 degree triangle

  This is the 30 degree special triangle. Across from the 30 degree angle is x, across from the 60 degree angle (not labeled) is x radical 3, and across from the 90 degree angle is 2x. The hypotenuse of this triangle is one.

  This pictures demonstrates a the sides and ordered pairs of the 30 degree special triangle. Since the the hypotenuse or the radius is equal to one, x (across from the 30 degree) has to be 1/2 and x radical 3 (across from 60 degree angle) has to be radical 3 over 2. The work is shown beneath the triangle. You can simply find the ordered pairs by looking at its points. This triangle is labeled at quadrant one on the graph. 

2. 45 degree triangle 

   This is a 45 degree special right triangle. Since two out of the three angles are 45 degrees, the letter x is the same. Across from the 45 degree is x and across from the 90 degree is x radical 2.

   This pictures reveals the ordered pairs and sides of the 45 degree special right triangle. Since the hypotenuse or radius is one, the letter x, across from the two 45 degrees, will be radical 2 over 2. The work is shown below the triangle. The triangle in this picture is located in quadrant one on the graph. Ordered pairs are plotted according to its location.

3. 60 degree triangle

   This is a 60 degree special right triangle. Across from 30 degrees (not labeled) is x, across from 60 degrees is x radical 3, and across from 90 degrees is 2x. This triangle is similar to the 30 degree special right triangle. However, you will noticed that the ordered pairs and sides will be switched.

   This picture shows the ordered pairs and sides of the 60 degree special triangle.  The hypotenuse/radius is one, which is similar to the last two special right triangles. The sides of this triangle is consisted of radical 2 over 2, 1/2, and 1. Work is shown below the triangle. Again, this triangle is labeled on the first quadrant. Ordered pairs are plotted according to its side and point.

4. This activity helps me derive the unit circle because it gives me the degrees, points, and ordered pairs that are located at quadrant one. Thus, with this knowledge, I can apply it to the other three quadrants on the graph. I can use the idea of reference angles to know what degrees  and ordered pairs (depending on the sign) will be on the unit circle for each quadrant. The special right triangle is part of the unit circle in a way that it depicts all the pieces that a unit circle has.

5. The triangles in this activity are located in quadrant one. Quadrant one is used to find all the other parts in the other quadrants. The "magic 5" relates to the other quadrants in terms of degrees, radians, and ordered pairs. The "magic 5" includes 0, 30, 45, 60, and 90 degrees. Knowing the first quadrant of the unit circle is very significant. Nevertheless, this is where the term reference angles and coterminal angles come in need. Coterminal angles are angles that have the same terminal side. Reference angles are positive, acute angles (from degree to x-axis).

http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_34.gif

  This pictures shows the 30 degree special right triangle in quadrant II, III, and IV. All of these three quadrants are similar to the quadrant I. The difference in these quadrants is the ordered pairs because of the signs depending where it's located. Each quadrant has the same reference angle, which is 30 degrees.

 

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/ebaa19ac-ff8b-43a6-a793-a00d9ac15e86.png

   This picture shows the 40 degree right triangle in all of the quadrants on the unit circle. The only difference in each quadrant is the sides/ordered pairs because they can either be a positive or negative depending where it is plotted. Reference angles are still the same.

http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif

   This pictures shows the 60 degree right triangle on the other quadrants (besides first quadrant). The sides are the same such as 1/2 still lies on the x-axis at quadrant III and IV. The difference, similar to the other special right triangles, is the sign on the ordered pairs. The values, otherwise, are still the same according to the first quadrant.

INQUIRY ACTIVITY REFLECTION

1. The coolest thing I learned from this activity was knowing how to relate special right triangles to unit circles.
2.  This activity will help me in this unit because I can now see the pattern and find a way to memorize all the degrees, radians, and ordered pairs in a unit circle.
3. Something I never realized before about special right triangles and the unit circle is how they are all connected to each other, in terms of the radius and the quadrants and all the other pieces that make up a unit circle.

References
http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/ebaa19ac-ff8b-43a6-a793-a00d9ac15e86.png
http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_34.gif
http://01.edu-cdn.com/files/static/learningexpressllc/9781576855966/The_Unit_Circle_21.gif

RWA #1: Unit M Concept 5: Graphing Eclipses Given Equation

1. Mathematical Definition of an Eclipse: "the set of all points such that the sum of the distance from two points is a constant (Kirch)"

2.   The equation for an eclipse is . To define it graphically, an eclipse should look like an oval-shaped. There are two types of eclipses, which is either a "fat" or a "skinny" eclipse. In a "fat" eclipse,  the major axis has a length of 2a and the minor axis has a length of 2b. Moreover, the major axis lies according to the y-value since it has to stretch out horizontally. In a "skinny" eclipse, the major axis lies according to the x-value since it has to stretch out vertically. Both the "fat" and "skinny" eclipses' minor axis are opposite to its major axis. Graphically, an eclipse consists of the center (h,k), major axis, minor axis, foci points, vertices, and co-vertices. The center is the intersection of the major and minor axis. The vertices are endpoints to a major axis, while the co-vertices are endpoints to a minor axis. The foci are the focus points to demonstrate how much the eclipse deviates from being circular.
   To find the parts of an eclipse, you can determine by solving or putting the puzzle pieces together algebraically. In standard form, the x-value will go with h, and the y-value will go with k. The term a^2 and b^2 can either be under the term x and y depending what type of eclipse it is. The standard form of an eclipse will always be added and equaled to one. If it is not equal to one, make sure to simplify it. The standard form give you clues whether an eclipse is skinny or fat, center, major axis, minor axis, a, and b. Algebraically, the major axis will always be the bigger number and the bigger number will always be known as the term "a". Once you found your major axis, it is easy to find your minor axis by looking at the standard form. Thus, the term for minor axis is "b". To find the foci points or "c", you can use the equation a^2 - b^2 = c^2. Also, finding the eccentricity is very significant because it gives you an exact number of an eclipse that deviates from being circular. You can find the eccentricity by taking "c" and dividing it by "a" or c/a. The eccentricity of an eclipse is 0<e<1.


This image depicts the two types of ellipses and the key points both graphically and algebraically.

Here's a video about ellipses!!


3.  Ellipses can be found in our universe such as the solar system! The way the planets orbit around the sun has a shape of an ellipse including the satellites. Astronomers called this movement as elliptical orbits. For instance, the Earth's movement around the Sun has an eccentricity of 0.0167. In the solar system, the Sun is the focus. Since the Sun is the focus of an eclipse, the planets can either move closer or further away from the sun.
   Sometimes when you stare out of the sky, you can see the moon with a solar ellipse. This is "when the lunar disk passes directly between us and the sun." A lunar ellipse happens when the moon is considered a full moon. "Because the Moon casts a smaller shadow than Earth does, eclipses of the Sun tightly constrain where you can see them. If the Moon completely hides the Sun, even for a moment, the eclipse is considered total." The latest lunar ellipse was in December 2011 in Los Angeles. 

4. References:
http://www.teacherschoice.com.au/images/ellipse_types.gif
http://www.mathopenref.com/images/coordgeneralellipse/Equation-3.jpg
http://www.teacherschoice.com.au/images/ellipse_types.gif
http://www.windows2universe.org/physical_science/physics/mechanics/orbit/ellipse.html
http://www.skyandtelescope.com/observing/highlights/237963491.html