I/D #3: Unit Q- Pythagorean Theorem

Inquiry Activity Summary: 
Where does sin^2x+cos^2x=1 come from?

The Pythagorean theorem is derived to become a Pythagorean identity. From the unit circle, the Pythagorean theorem uses "x", "y", and "r" which would become: x^2+y^2=r^2. With this we can perform an operation which allows the equation equal to 1. The operation is to just simply divide r^2 from both sides of the equation which in turn creates (x/r)^2+ (y/r)^2= 1. This then becomes an identity, proven facts and formulas that are always true. From the unit circle we see that cosine equals x/r and sine equals y/r, with these two we can just substitute them into the equation which will then give us the Pythagorean Identity sin^2x+cos^2x=1.

Show and explain how to derive the two remaining Pythagorean Identities from sin^2x+cos^2x=1.

Please look at the pictures below, they already have the descriptions on how I got other two remaining Pythagorean Identities.

Inquiry Activity Reflection:

The connections that I see between Units N, O, P, and Q so far are that they all relate back to the Unit circle. The reason why I say this is because of the fact that when we derive triangles, Heron's Theory, and any of the other equations that we have learned prior to this would reference back to the Unit circle for the legs and hypotenuse which will help us to derive the triangle or theory.

If I had to describe trigonometry in three words, they would be complicated, interesting, and enjoyable. It's complicated because we learn not only the equations and theories given to us but we also learn how it was derived and how it is connected with the real world. It is interesting because prior to learning these mathematical equations and theories you don't really question the math that is going on around you and you especially don't think about how these formulas were derived, so it makes it interesting to see all the math that is around you everyday. And finally it is enjoyable because when you finally understand how to do the problems and understand how you are connected to it, you start to enjoy answering the problem and you begin to see that math isn't as bad as many may think it is.

WPP 13 & 14: Unit P Concepts 6-7

This WPP 13-14 was made in collaboration with Genesis R.. Please visit the other awesome posts on her blog here.

Hershey has stopped at a stoplight and noticed that his best friend, Marlene, is due west of him at the next stoplight, 30 feet away. Both are going to Hershey's Bakery. Hershey walks N 30* W to get there while Marlene goes N 72* E to get there. What is both of their distances to walk over to Hershey's Bakery?

After having a nice, long conversation with Hershey, they decided to see each other again the next day. They both leave the bakery at the same time. Marlene, in hurry to get to her cousin's Quinceanera, is headed at a bearing of 315* and is traveling 50 MPH. Hershey on the other hand goes home at 30 MPH at a bearing of 078*. How far apart are they after two hours?

BQ #1- Unit P: Concepts 1&4: Law of Sines AAS or ASA and Area of an Oblique Triangle

i-Law of Sines:
Why do we need it?
The law of sines is of importance because it helps us to find non-right triangles that are commonly seen in the real world. With this we are able to find any unknown angle or side if we have already found or were given two angles and one side as well we could have been given two sides and one angle.
How is it derived from what we already know?

http://i1.ytimg.com/vi/Bj7h6OMBvqk/hqdefault.jpg

 When given an unknown triangle with some given information on the side and two angles, when deriving it we can just form a line straight down from the middle to give us two 90 degree triangles. With this we can use Sine and it will  be something as below where we get a ratio. However since we want Sine by itself we multiply through to receive the final answer.

http://www.lhs.loganschools.org/~rweeks/trig/law_of_sines.jpg

iv-Area Formulas:
How is the "area of an oblique" triangle derived?
 When given a triangle with some of the given angles and side we just cut the triangle in half for we can receive two 90 degree triangles. With this we then use SOHCAHTOA to get a ratio, with this we can substitute it in for the height of the area formula we are used to, for we can use to solve the area of the triangle.

https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRUfaYjkTy5hlsAuerPsDTfulo_zx2qECBMGIWGwX-EcYy4zuaMlV3LC1XqmMsvMpN1kXNTXUSw9HJScLKdDdZdydgqvLghasCFxtONLYiSbtPDbOBMUe9xWWI1TqvDtxuW0-MR7RxxZY/s400/hi.bmp

How does it relate to the area formula that you are familiar with?
This relates to the area formula (a=1/2bh) that we are familiar with because we solve with this formula however we substitute "h" with 1/2abSinC or any of the others.

References:
http://www.lhs.loganschools.org/~rweeks/trig/law_of_sines.jpg
http://facstaff.gpc.edu/~ahendric/Math1113/sec6_1notes/images/pic010.jpg
http://i1.ytimg.com/vi/Bj7h6OMBvqk/hqdefault.jpg
https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRUfaYjkTy5hlsAuerPsDTfulo_zx2qECBMGIWGwX-EcYy4zuaMlV3LC1XqmMsvMpN1kXNTXUSw9HJScLKdDdZdydgqvLghasCFxtONLYiSbtPDbOBMUe9xWWI1TqvDtxuW0-MR7RxxZY/s400/hi.bmp

WPP #12: Unit O: Concept 10- Solving angle of evelvation and depression word problems

Create your own Playlist on LessonPaths!

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

1-"Something that I never noticed before about special right triangles is ..." that when you use other numbers other than the ones given you get a similar answer as you did with the numbers given to you.
2-"Being able to derive these patterns myself aids in my learning because ..." it helps to know how you got these equation for the special right triangles and it helps you learn what you are doing.

I/D #1 Unit N: Concepts 7-9: How do you SRT and UC relate?

Inquiry Activity Summary:
Please watch the videos below because they will help you in understanding from where we get our points for the unit circle and they will show you how the special right triangles relate to the unit circle:

This video helps you to derive from the unit circle in how it is a thirty degree angle and is one of the often used angle and it helps to know how the point for this degree on the unit circle came from. When the triangle is drawn in a different quadrant not only does the degree change but the x or y values can change into a negative.

This video helps to derive from the unit circle in how it is the forty-five degree angle and it is able to help us know how we got the point for this degree on the unit circle. When the triangle is drawn in a different quadrant the x or y can become a negative or the both can become a negative.

This video helps to derive from the unit circle in how it is able to help us learn in how we got the points for this degree. When the triangle is in another quadrant both the x and y can become negative but the point given to us will still be the same.

Inquiry Activity Reflection:
 The coolest thing I learned from this activity was how we actually got the points and all the math that actually happens behind the unit circle.
 This activity helped me on this unit because it made it a lot easier for me to understand and memorize the unit circle.
 Something I never realized before about the special right triangles and the unit circle is the fact they related to one another and that they make it a lot more easier to figure out what is the points of these angles.

RWA #1: Unit M Concepts 4-6: Conic Sections in Real Life

Parabolas:
1-"The set of all points that are equidistant from a point that is known as the focus and a line known as the directrix." (Mrs.Kirch's LessonPaths Playlist Unit M Concept 4a)
2- The equation for a parabola is (x-h)^2=4p(y-k) or (y-k)^2=4p(x-h), with this we are able to receive pieces of information and will help to graph out the needed information.

http://www.mathwords.com/p/p_assets/parabola%20features%20focus%20directrix%20vertex%20axis.gif

     Some of the key features for the graphing of this conic section is the fact of seeing whether it goes up, down, left or right. Well to figure this out you must look to see which goes first/is squared and by this I mean the "x" and "y" in the equation that was given in the first sentence above the picture. With this information you will be able to learn whether it goes up/down or left/right as well will you be able to identify the center/vertex is by using the "h" and "k" that is given in the equation. However it will need "p" for you can further know if it is going to the right or down, "p" is the space between the focus and the vertex as well as the distance between the vertex and directrix. With this in mind, "p" has a huge impact on the hyperbola for it can show us whether the graph will be thin or wide but it also shows us the slope for the graph.
      There is also the directrix, which is given or you must find, and this helps to show you were your parabola must go over or next to. With the directrix you get the general idea of how the graph should look like at the end. There is also an axis of symmetry that goes through the focus and vertex and helps us to see in which direction the parabola should go into. However if still confused here is a video that may help you to further understand hyperbolas.

3- What it can be in the real world (example):

http://i00.i.aliimg.com/img/pb/179/937/366/366937179_304.jpg

      A real world application of a hyperbola may be a jet of water, for better definition I mean the formation of the water of a fountain. What I mean is that when the water is going upward gravity is still pulling down making the water curve making it look like a parabola.
      If this was graphed out it would have a directrix right above the curving point, in which we can consider as the vertex, and a focus that will show us the approximation of how wide the water will be when falling back into the fountain. If you want to know more or just want to see more of what parabolas are in real life go here: http://www3.ul.ie/~rynnet/swconics/UP.htm

 4- Work cited:
"Applications of Hyperbolas." Applications of Hyperbolas. N.p., n.d. Web. 09 Feb. 2014.
  "Equation of a Parabola (conic Section)." YouTube. YouTube, 13 Mar. 2013. Web. 09 Feb. 2014
 "This Learning Playlist Is Empty." LessonPaths. N.p., n.d. Web. 11 Feb. 2014.
 http://i00.i.aliimg.com/img/pb/179/937/366/366937179_304.jpg
http://www.mathwords.com/p/p_assets/parabola%20features%20focus%20directrix%20vertex%20axis.gif