Wednesday, March 26, 2014

I/D#3: Unit Q: Pythagoreom Identities

Inquiry Activity Summary: 
1.Sin2x+cos2x=1 comes from the identity, Pythagorean theorem, which is a  factor or formula that is proven.The Pythagorean theorem, a^2+b^2=c^2 is replaced x^2+y^2=r^2 because of the fact that  they hold the same values.

2. To derive the two remaining Pythagorean Identities from sin2x+cos2x=1 you must first, divide everything by r^2 and after doing that we get x^2/r^2+y^2/r^2=1. You can then notice that they x/ r is cosine and y/ r is sine meaning that you can put sin2x+cos2x=1 in place.

Inquiry Activity Reflection:

1. The connections that I see between Unit N,O,P, and Q so far are all related to the unit circle. In order to solve and understand most of them, you must fully understand and keep in  mind the unit circle. Having memorized the unit circle is key. 

2. If i had to describe trigonometry in THREE words, they would be difficult, challenging, and memorization. Trigonometry really requires a lot of practice and memorization of the unit circle. You can not solve the problems if you do not know the unit circle, formulas,  sin, cos, tan, and so much more. 

Monday, March 17, 2014

BQ#1: Unit P Concepts 2 and 4: Law of Sins and Area Formulas

2. Law of Sins: Why is SSA ambiguous?
It is ambiguous because sometimes the three parts that are given will one triangle, two triangles, or not not one triangle at all. Below is an example of a two triangle.

4. Area Formulas: How is the “area of an oblique” triangle derived?  
It is derived from the triangle area formula= 1/2bh. In order to know the height of we have to make two triangles by drawing a line down the middle from angle B. We then label that line h because it is the height. With the given information above in the picture, we take the sin of A and equal it to Sin C. We multiply C to each side leaving us with CsinM<A=h. Next, plug in 1/2bh and we get A=1/2b(asinC). For a better understanding click the link below watch the video:

Wednesday, March 5, 2014

WPP#12: Unit O Concept 10: Solving Elevation and Depression Problems
1A) Since Ana is not an early bird, Ricky decides to get a head start on his climb to the top of Creek Mountain without her. He is 100 ft above ground level while Ana is still in her tent 60 feet from the base of the mountain. What is the angle of elevation in which Ana's eyes will see Ricky climbing the mountain?

1B) Ana finally decides to wake up to catch up to Ricky and they both then climb to the top of the mountain which is 215 ft above ground level. They are so high above that they can see their tent from a 30 degree angle of depression. What is the horizontal distance above ground from the top of the mountain to the tent?


Tuesday, March 4, 2014

I/D#2: Unit O- How Can We Derive the Patterns for a Special Right Triangle?

1. 30-60-90 Triangle
This equilateral triangle has all sides equalling one (because given) and the angles equalling 60 degrees. We know that because all angles in a triangle sum up to 180, and since there are 3 angles you divide 180 by 3 giving you 60 for each angle. However, to get our 30-60-90 triangle, we must cut the triangle vertically in half, having the top 60 degrees become two 30 degree angles and the middle bottom of the triangle two 90 degree angles as seen below. As for the sides equalling one, since we cut the triangle in half, each triangle is going to have the bottom side equalling 1/2. To find the other sides, we use the Pythagorean theorem which is a^2+b^2=c^2. In this case were solving 1/2^2+b^2=c^2 which results in b= radical 3 over 2. To make it easier and not deal with fractions we multiply everything by 2 and add a variable, n, to represent any number, meaning the relationship is consistent. We then get a=2n , b=radical 3 over 3, and c= n.

2. 45-45-90 Triangle
Since we want a 45-45-90 triangle, we cut the square diagnolly in half. How do we know we got two 45 degree angles? Well, a square has all 90 degree angles and when we cut it diagonally in half, the two 90 degree angles split into two, meaning the 90 degree angle is divided by 2 as seen below. We are given that all sides equal one but since we don't not know one of the sides because we cut the square we use the Pythagorean theorem which is a^2+b^2=c^2. In this case were solving 1^2+1^2=c^2, giving us c=radical 2. We then add n to each side in which represents any number meaning the relationship is consistent.

1. Something i never noticed about a special right triangle is that there will always be two sides that are the same for example a 45-45-90 triangle a and b are the same and in a 30-60-90 triangle.
 2. Being able to derive these patterns myself aids in my learning because it gives me a better understanding of how everything connects. If i every forget the rules i can simply derive them.