I/D# 1: Unit N Concept 7: How do special right triangles and the unit circle compare?

Deriving the Unit Circle

Inquiry Activity Summary:

1. 

Before Reduction (http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_30_60_90.gif)

A 30° Triangle has three sides to it: adjacent (x), opposite (y), and hypotenuse (r). The opposite side is always x. The adjacent will be xradical3 and the hypotenuse will be 2x. To derive the unit circle from this triangle, the hypotenuse must equal 1. To get that to happen, all three sides have to be divided by 2x and simplified. After reduction the values become 1 for hypotenuse, radical3/2 for adjacent, and 1/2 for opposite. The resulting sides can be plotted and the coordinate near the 60° measure will be the coordinate used in the unit circle. The pair of coordinates can be used for the 30° angle measure and all its supplemental angles (150°, 210°, 330°). For the 30° degree triangle we know the coordinate is (radical3, 1/2). An example of all the steps being taken can be found in the bottom:

2. 

Triangle before reduction (http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_45_45_90.gif)

A 45° Triangle is composed of two sides that are the same length (x and y) and a hypotenuse. The equal sides are labeled as x and the hypotenuse is xradical 2. To derive the unit circle values, we must get the hypotenuse to equal 1. By dividing every value by xradical2 we can accomplish this and simplify to get our distance values that can be plotted out. The final values after reducing are 1 for hypotenuse, and radical2/2 for both adjacent and opposite. The resulting coordinate near the 45° angle measure will be used for 45° in the circle and all its supplemental angles (135°, 225°, 315°). That value is (xradical2, xradical2). An example of this derivation can be seen in the following image:

3. 

Triangle before reduction (http://img.sparknotes.com/figures/B/b21b8be4e663f46072a3c3b19b657891/306090.gif)

A 60° Triangle has the same sides as a 30° triangle. The opposite side is always xradical3. The adjacent will be x and the hypotenuse will be 2x. Since the location of the degrees change, the same simplifications are taken but they end up in different values. The final measures after reducing are 1 for hypotenuse, 1/2 for adjacent, and radical3/2 for opposite. The coordinate pair near the 30 measure will be used in the unit circle, the pair is (1/2, radical3/2). This value is the same for 60° and all its supplementary angles (120°, 240°, 300°). An example can be found in this image:

4.

This activity helps me derive the Unit Circle because the triangles reflect in the circle throughout all four quadrants. If reflected upon the y-axis, the coordinates measure the same from the bottom up. If reflected from there down the x-axis, the coordinates measure the same from the top down. If shifted from the first quadrant down across the x-axis, the coordinates measure the same from the top down. The only things that change are the signs and those sign changes can are explained in section 5.

5.The triangle in this activity lies in quadrant I since both x y values in the (x,y) coordinates are positive. If you change quadrants, the coordinate values would change. If you drew the triangles in quadrant II, the x value would become negative and the y would remain positive. If you drew the triangles in quadrant III both the x and y values would become negative. If you drew the triangles in quadrant IV, the x value would remain positive and the y values would change to negative. To view an example of different quadrants and the change in coordinates that occur along with them view this image derived by myself from repeating triangles:

Inquiry Activity Reflection:

1. The coolest thing I learned from this activity was how easy it is to follow the triangles throughout the different quadrants.

2. This activity will help me in this unit because when I feel lost if I forget my coordinates or angle measures, I can always use the first five to figure out the rest of the unit circle or try to derive the triangles again if my mind if blank.

3. Something I never realized before about special right triangles and the unit circle is that the values in the unit circle correlate with the values found when the special right triangles' hypotenuse is reduced to one.

References:

30 degree triangle image: http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_30_60_90.gif

60 degree triangle image:http://img.sparknotes.com/figures/B/b21b8be4e663f46072a3c3b19b657891/306090.gif

45 degree triangle image:http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphics/trig_45_45_90.gif

RWA# 1: Unit M Concept 6: Hyperbola conic section in real life

All There is to Know About Hyperbolas

1. Mathematical Definition: A hyperbola is "the set of all points such that the difference of the distances between each of two fixed points (the foci) and any point on the hyperbola is a constant" (Mrs. Kirch)

2. Describing the Conic Section

Algebraically:

Formula of a Hyperbola (Fig. 1) (http://www.mathwarehouse.com/hyperbola/images/compare-hyperbola-graphs.gif)

The formula of a hyperbola varies upon which way its "branches" (or curves) branch out. If its formula begins with x, it will open up on the x axis and its transverse axis will thus be "y=a value". If it begins with y, it will open up on the y axis and its transverse axis will thus be "x=a value"(Fig 1). On this axis will lie the hyperbola's center, vertices, and foci. The denominators below the x and y squared denominators will direct how many units out the hyperbola's axes will go from the center (the center of a hyperbola is defined as (h,k)). Those values are defined by "a" and "b". Whichever axis does not follow the direction in which the hyperbola opens is called the conjugate axis. On this axis will lie the hyperbola's co-vertices. 

Asymptotes are in the form of "y=mx+b" and they are formed by the parameters set by the "a" and "b" values. Asymptotes give the hyperbola its shape and can be calculated using the equation shown in Figure 2

Asymptotes of a hyperbola (Fig. 2) (Mrs. Kirch SSS)

To find the values of "a", "b", and "c", the formula in Figure 3 is used. "a" is how many units should be distanced from the center to form the vertices. "b" is distance from the center that form the co-vertices. "c" is the distance from the center that mark the foci.

(http://nado.znate.ru/images/ukbase_2_1534681587_469.jpg) Fig. 3

Image showing the formula of a hyperbola and some of its properties:

(http://www.mathwarehouse.com/hyperbola/images/compare-hyperbola-graphs.gif) Equation of a hyperbola and some key features (Fig. 5)

Graphically:

Visually, a hyperbola is formed by the curve that is formed when "the plane cuts through both nappes [tips] of the cone." (About Mathematics) 

(http://mathworld.wolfram.com/images/eps-gif/ConicSection_1000.gif) Hyperbola as a conic section (Fig.4)

A hyperbola is composed of two axes, its transverse and conjugate axis. On the transverse lie the center, vertices, and foci. The transverse axis can be vertical or horizontal, depending on the hyperbola's equation (see "Algebraically" section) On the conjugate lie its co-vertices. The center is defined by (h,k) and the vertex is determined by "a" units from the center. The conjugate axis can be vertical or horizontal, depending on the equation (see "Algebraically" section). The co-vertices are determined by going "b" units form the center in the direction of the conjugate axis. The "rectangle" formed by the distances traveled by "a" and "b" will outline a parameter which is crossed by the hyperbola's asymptotes. The asymptotes determine the shape of a hyperbola and how wide or skinny it is.  The video below shows the properties of a hyperbola algebraically and graphically.

Foci effects on the shape of the hyperbola and its eccentricity: 
The eccentricity for an ideal hyperbola should be greater than one. The closer the eccentricity is to one, the "pointier" it will be. The farthest it is greater than 1, the more spaced out, "fat", it will be. Refer to Figure 6 for an example of eccentricity and its effect on a graph's shape. The formula c over a represents eccentricity.

(http://www.purplemath.com/modules/hyperbola.htm) Eccentricity effects on graph (Figure 6)

A shorter foci will mean the hyperbola will be smaller in terms of wideness. The farther out the foci are, the wider the hyperbola will be in the long run. To play around with hyperbola foci and their effects on eccentricity and shape, feel free to visit this interactive tool

For a  detailed website that gives a very clear outline about the looks of different eccentricities and how to foci can effect it click here.

3. Hyperbolas in real life:

             Hyperbolas play parts in our lives that one might not notice without looking into it. Comets follow paths in shapes of a hyperbola, telescopes are made out of hyperbolic lenses, sound waves travel in paths that resemble hyperbolas, sonic boom shock waves are shaped in a hyperbola shape, and there are many more other real life applications. One of the most useful applications for hyperbolas is in radio waves, more specifically, in LORANS (long-range navigation systems). When radio signals are emitted from two points, they "form concentric circles intersecting each other" (lessonpaths playlist). The patterns that result by the meeting circles create hyperbolas. This is the basis for the navigation system.

Use of hyperbolas in LORAN geographic calculations (Fig7) (http://www.ibiblio.org/hyperwar/USN/ref/RADTWOA/img/fig5-1.jpg)

           LORANS use time differences between the radio signal reports from two differently located stations two find the location of the receiver on the hyperbola. To get more accurate results, a third station might be used to find the exact location of the receiver. The United States developed this system after modeling it off of a British system used in WW2, according to a powerpoint on conic secitons in real life (lessonpaths playlist). The system was formerly known as LRN (Loomis radio navigation) because it was named after the physicist who invented it, Alfred Loomis. Most widely used by the US Navy, the earliest forms of LORAN systems had ranges of up to 1,200 miles. LORAN was the most popular form of navigation until GPS systems were created. 

*Most information gathered from this source.

References: 

About Mathematics: http://math.about.com/library/blconic.htm (Definition of a hyperbola)

Hyperbola formula and its sections image: http://www.mathwarehouse.com/hyperbola/images/compare-hyperbola-graphs.gif (Figure 5)

Hyperbola conic section: (http://mathworld.wolfram.com/images/eps-gif/ConicSection_1000.gif) (Figure 4)

Unit M SSS Asymptote Equations: https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkb2NzZm9ya2lyY2hzY2xhc3Nlc3xneDo0MGEzODAwOWE3NjAxOWE (Figure 2)

Hyperbola algebraic and graphic properties video: http://www.youtube.com/watch?v=kLvcELrlT4U

Foci and its effects interactive tool: http://www.lessonpaths.com/learn/i/unit-m-conic-section-applets/hyperbola-from-the-definition-geogebra-dynamic-worksheet 

Very detailed website about the hyperbola and all its properties (hyperlinked website) : http://www.purplemath.com/modules/hyperbola.htm

Hyperbolas in real life (lessonpaths hyperlinked playlist): http://www.lessonpaths.com/learn/i/unit-m-conic-sections-in-real-life/conic-sections-in-real-life

LORAN map image: http://www.ibiblio.org/hyperwar/USN/ref/RADTWOA/img/fig5-1.jpg (Figure 7)