Saturday, February 22, 2014

I/D 1 Unit N Concepts

1. A triangle with one right angle is called a right triangle. The side opposite the right angle is called the hypotenuse of the triangle. The other two sides are called legs. The other two angles have no special name, but they are always complementary. Do you see why? The total angle sum of a triangle is 180 degrees, and the right angle is 90 degrees, so the other two must sum to 90 degrees. One is the right triangle formed when an altitude is drawn from a vertex of an equilateral triangle, forming two congruent right triangles. The angles of the triangle will be 30, 60, and 90 degrees, giving the triangle its name: 30-60-90 triangle. The ratio of side lengths in such triangles is always the same: if the leg opposite the 30 degree angle is of length x , the leg opposite the 60 degree angle will be of x , and the hypotenuse across from the right angle will be 2x . Here is a 30-60-90 triangle pictured below. Which side is which? The side opposite the 30 degree angle will have the shortest length. The side opposite the 60 degree angle will be sqrt(3) times as long, and the side opposite the 90 degree angle will be twice as long. The triangle below diagrams this relationship.

30-60-90 triangle side ratios proof: Proving the ratios between the sides of a 30-60-90 triangle


2. 45-45-90 triangle side ratios: Showing the ratios of the sides of a 45-45-90 triangle are 1:1:sqrt(2)


The other common right triangle results from the pair of triangles created when a diagonal divides a square into two triangles. Each of these triangles is congruent, and has angles of measures 45, 45, and 90 degrees. If the legs opposite the 45 degree angles are of length x , the hypotenuse has a length of x . This ratio holds true for all 45-45-90 triangles. 45-45-90 triangles are also often called isosceles right triangles.


4.This Activity help us derive the unit circle by knowing the lengths . of we forget something about the unit circle

Sunday, February 9, 2014

RWA #1 Unit M Concept 5 elipses

1. Ellipse - The set of all points such that the sum of the distance from two points is a constant.

The ellipse


2.  To find the ellipse it has to be in standard conic form which is (x-h)^2/a^2+ (y-k)^2/b^2 = 1. It doesn't matter in what order a^2 and b^2 are they can be under whichever equation. h and k have to be in a specific one. H always has to be with x and k always has to be with y. another thing is that to determine the size of the ellipse if it's skinny or fat has to be based on x or y. If there ia a bigger number under x then it's fat, but if it's under the y then it's skinny. As you can see on the picture whatever d and d 2 are when they are added they equal 2a.
    When graphed h,k will be center points. What ever numbers are on the bottom like we said then it  place will be a and the second one b.vertices come from a, co-vertices from b, and foci from c and we find c from a^2-b^2=c^2. If a has bigger numbers then it's our major axis and b will be minor axis. The foci will be on the major axis but a little before the vertices points. To determine the eccentricity of an ellipse,is e = c/a. So it's whatever number is c comes out to be divided by whatever number a is. If it is still not clear the video on youtube below this will end up explaining how to solve the problem.


3.You can find ellipses in a training machine. "An elliptical training machine simulates the motion of running or climbing to provide the user with a healthful exercise without any impact on the joints." this helps the people to have better results little by little.
  "The foot of a user describes the shape of an ellipse as the machine is used. An elliptical machine can be motor-driven or user-driven, as well as dual action, where handlebars and leg mounts interdependently provide motion for each other. The elliptical trainer provides a stationary exercise for those who wish to avoid joint injury as a result of running."
( this ellipse figure really helps the people that are trying to avid injuries so ellipses have a big impact on that.