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

1. 30 Triangle 

The triangle is at quadrant one and label the three sides. Hypotenuse is 2x, adjacent is x radical 3, and opposite is x. To get the values we will need to derive them. For hypotenuse divide by 2x to get one (r value), for adjacent divide by 2x to get radical 3/2 (x value), for opposite divide by 2x to get 1/2 (y value). Finally draw the coordinate planes (0,0) as the orgin, (radical 3/2,0) as the x axis, (radical 3/2,1/2) as the y axis. 

30 triangle

2. 45 triangle

 The triangle is at quadrant one and label the three sides. Hypotenuse is x radical 2, adjacent is x, and opposite is x. Then derive to get the values. For hypotenuse(r) divide by x radical 2 to get one, for adjacent(x) divide by x radical 2 to get 1/radical 2 then rationalize to get radical 2/2, and for opposite(y) divide by x radical 2 to get  1/radical 2 then rationalize to get radical 2/2. Finally draw the coordinates planes, (0,0) as the origin, (radical 2/2) as both the y and x axis. 

45 triangle

3. 60 triangle 

The triangle is at quadrant one and label the three sides. Hypotenuse is 2x, adjacent is x, and opposite is x radical 3. Then derive to get the values. For hypotenuse(r) divide by 2x to get one, for adjacent(x) divide by 2x to get 1/2, and for opposite(y) divide by 2x to get radical 3/2. Finally draw the coordinate planes, (0,0) as the origin, (1/2,0) as the x axis, and (1/2,radical 3/2) as the y axis. 

60 triangle

How does this activity help you to derive the Unit Circle?

This activity helps me derive values that are in the unit circle. Also the values are either positive or negative depending on what quadrant it lies on. All the values and trig functions in quadrant one are positive, in quadrant two the x axis is negative and sine and cosecant are positive, in quadrant three both x and y axis are negative and tangent and cotangent are positive, and for quadrant four the y axis is negative.

THE COOLEST THING I LEARNED FROM THIS ACTIVITY WAS that the triangles and its coordinate planes make up the entire unit circle. Also when dividing with the value of hypotenuse on the adjacent and opposite sides gets us the values from the unit circle. 

THIS ACTIVITY WILL HELP ME IN THIS UNIT BECAUSE now I know what order pair goes in what degree all thanks to the 30, 45, 60 triangles. Also where the values and trig functions are positive or negative. 

SOMETHING I NEVER REALIZED BEFORE ABOUT SPECIAL RIGHT TRIANGLES AND THE UNIT CIRCLE IS that deriving the triangles to get the same values from the unit circle would be easy get. Also that the unit circle was made from the 30, 45, 60 triangles. 

I/D#3: unit Q-Pythagorean Identities

• First of all an identity is a proven fact that is always true. The Pythagorean Theorem when using x,y and r, it's x^2+y^2=r^2. Then divide everything by r^2 to get one. It will look like this (x^2/r^2)+(y^2/r^2)=1. Since cosine is x/r and sine is y/r. So it will be replace to be cos^2x+sin^2x=1. That's why it's referred to the Pythagorean Theorem. The magic 5 pairs to make an identity true are (1/2,3/2),(2/2,2/2),(3/2,1/2), (0,1) and (1,0). The connection between units N, O, P, and Q so far are the magic 5 pairs or the unit circle and using sin, cos and tan as well as its reciprocals. If I had to describe trigonometry in THREE words, they would be complicated, long and strategically to solve. 
• 
  
• For further understanding visit this websites, number three is a video explaining the process:
• 1.http://math.tutorvista.com/geometry/pythagorean-identities.html?view=simple
• 
  
• 2.http://www.purplemath.com/modules/idents.htm
• 
  
• 3.http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-27-trig-integrals/

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

For the 30-60-30 triangle label all the sides by 1. Divide the triangle in two. That will divide the 60 degrees in to 30 degrees, the adjacent in half and make a 90 degree. Use the Pythagorean theorem to get the opposite side. Multiply by 2 to leave the radical alone and get 1 in the adjacent side and get 2 in the hypotenuse side. 

 For a 45-45-90 triangle label all the sides by 1. Divide the square to get two triangles. That will divide the 90 degrees into 45 degrees. Use the the Pythagorean theorem to get the hypotenuse side.  

 We use "n" to reference the missing value.

Something I never noticed before about special right triangles is, why we use "n" for?

Being able to derive these patterns myself aids in my learning because it gives a picture of both triangles and its formula. 

This are pictures of both triangles and showing how to get their formulas.

30-60-30 triangle

45-45-90 triangle