I/D# 1: Unit N: How Do Special Right Triangles and the Unit Circle Relate?

Inquiry Summary Activity

1.) This is a 30º triangle.

As you can see, the properties of a 30º SRT are that the side opposite the angle has a value of x, the side adjacent to the angle has a value of x radical 3, and the hypotenuse has a value of 2x.

 

We simplify the hypotenuse to make it equal to 1, but whatever we do to one side must be done to the other sides. When we divide the hypotenuse by 2x, we also divide the opposite and adjacent sides by 2x, which reduce to 1/2 and radical 3 over 2 respectively.

 

 

 

When we place the triangle on a coordinate plane so that the angle is at the origin, we can figure out the coordinates of the points of the triangle. The point with the 30º angle is at (0,0), the point with the right angle is at (radical 3 over 2, 0), and the last point is at (radical 3 over 2, 1/2).

 

2.) This is a 45º triangle.

 

As you can see, the properties of a 45º SRT are that the hypotenuse has a value of x radical 2 and the other two sides have a value of x.

 

We simplify the hypotenuse to make it equal to 1, but whatever we do to one side must be done to the other sides. When we divide the hypotenuse by x radical 2, we also divide the other sides by x radical 2, which both reduce to radical 2 over two.

 

 

 

When we place the triangle on a coordinate plane so that the angle is at the origin, we can figure out the coordinates of the points of the triangle. The point with the 45º angle is at (0,0), the point with the right angle is at (radical 2 over 2, 0), and the last point is at (radical 2 over 2, radical 2 over 2).

 

3.) This is a 60º triangle.

As you can see, the properties of a 60º SRT are that the side opposite the angle has a value of x radical 3, the side adjacent to the angle has a value of x, and the hypotenuse has a value of 2x.

 

 We simplify the hypotenuse to make it equal to 1, but whatever we do to one side must be done to the other sides. When we divide the hypotenuse by 2x, we also divide the opposite and adjacent sides by 2x, which reduce to radical 3 over 2 and 1/2 respectively.

 

 

When we place the triangle on a coordinate plane so that the angle is at the origin, we can figure out the coordinates of the points of the triangle. The point with the 60º angle is at (0,0), the point with the right angle is at (1/2, 0), and the last point is at (1/2, radical 3 over 2).

 

4.) This activity helps us derive the Unit Circle by showing us how to get the ordered pairs that make up the circle. When we make the hypotenuse equal to 1, we essentially set it as the radius of the circle. We also now understand which ordered pairs are connected with which SRTs.

 

5.) For this next section, we will be focusing specifically on the coordinates of the radius- or rather, how they change when the SRTs are moved to different quadrants.

 

 

When we move the 30º triangle to the second quadrant, the y value of the ordered pair stays the same, but the x value is now negative because it is on the negative side of the x axis.

 

 

When we move the 45º triangle to the second quadrant, the x value is still negative because it is on the negative side of the x axis, and the y value is now also negative because it is on the negative side of the y axis as well.

 

When we move the 60º triangle to the second quadrant, the x value is positive again because it is back on the positive side of the x axis, and the y value is still negative because it is still on the negative side of the y axis.

 

Inquiry Activity Reflection

 

1.)  “The coolest thing I learned from this activity was" how SRTs can be used to find the 'Magic 3' ordered pairs of a Unit Circle.

 

2.)  “This activity will help me in this unit because…” I now know how to derive the ordered pairs of the UC, so now I don't have to rely on my subpar memory when drawing it out.

 

3.) “Something I never realized before about special right triangles and the unit circle is…” the properties of the SRTs actually play a big part in deriving the ordered pairs of the UC.

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

1. Definition
An ellipse is the set of all points such that the sum of two points called foci equals a constant.

2. Description
The equation of an ellipse is as follows:

(x-h)^2 + (y-k)^2

   a^2          b^2

The x and y variables represent the coordinates of a point on the graph of the ellipse. The h and k variables represent the x value and the y value of the center, respectively. The a and b values determine the distance of the side of the ellipse from the center along the major and minor axes, respectively. The length of the major axis is double the value of a, and the length of the minor axis is double the value of b. To differentiate between a and b, you must keep in mind that a will always be the larger number, regardless of whether it is under the y term or the x term.

The graph of an ellipse is as follows:

http://www.csgnetwork.com/ellipse.png

As you can see, there are more variables that cannot be known just by looking at the equation. The c value is the distance from the center to the foci. It is always located along the major axis. To find c, use the equation a^2 - b^2 = c^2. The foci are two points that are also along the major axis. When you connect the foci to another point on the ellipse (see the blue), the sum will always be a constant. Additionally, the closer the foci are to the center, the more circular the ellipse will be. This value of how much the conic section deviates from being circular is called eccentricity. To find eccentricity, use the equation c/a.

3. Real World Application
One special property of an ellipse is that energy or sound directed at one focus will reflect to the other focus. A real world application of this property is called a whispering gallery. One person can stand at the focus of the whispering gallery and say something. However, whatever they say will not be heard by anyone except someone who is at the other focus. The sound waves bounce off the sides of the ellipse and reflect to the other focus.

A video of how the foci of an ellipse are related is as follows:

As the video shows, the sum of the distance between the two foci and the two points stays constant throughout the formation of the ellipse. A sound wave travelling a certain distance through one of the foci and bouncing off the side of the wall is like making the first segment. Since it can only go a certain distance more, it must reflect toward and through the other focus.

4. Works Cited
Image of an ellipse found here.
Definition of a whispering gallery found here.
Video link for the construction of an ellipse found here.