Welcome to William's Math Analysis Blog

Monday, March 24, 2014

ID#3 Unit Q: Concept 1 : Pythagorean Identites

Where does where sin2x+cos2x=1 come from?
First we have to remember that the Pythagorean theorem is an identity, and what this means is that no matter what the formula will always be true. When we use x, y, and r means that we are basing things on the unit circle. for this reason the identity is equal to 1, because we are basing ourselves on how the unit circle always equals to one. we know that the ratio of cosine in the unit circle is x/r and the ratio of sine in the unit circle is y/r. What i notice is that everything of this is related to the unit circle and the ratios in it. for that reason is that the identity of sin2x+cos2x=1. 

Show and explain how to derive the two remaining Pythagorean Identities from sin2x+cos2x=1
One of the first things that you should do is to decide which trig function you want to derive. if it's sin firs you have to move sine to the other side and do the same with the one ending up with 1 - cos2x = sin2x. and you do the same thing for cosine ending up with 1 - sin2x = cos2x. 

 The connections that I see between Units N, O, P, and Q so far are that all the trigonometric functions in the end connect with each other. They all come back to its origins which for us was the unit circle. 

If I had to describe trigonometry in THREE words, they would be... Relationship, Angles, Confusing

BQ# 1: Unit P Concept 1,2, and 4: Law of Sines, Area of Oblique Triangle

The Law of Sines
First of all this law is used whenever we get a non-right triangle in which the normal Trigonometric functions can't be used. This is used to find angles and sides of triangles with different angles and sides. The triangle should be labeled A, B, and C for the angles. Then after that we should label the sides with the same lower case letters, labeling the side opposite to the angle. You must divide the triangle into two creating a line representing the height and with that we can start to work with sines. From this point we can get two different equations. SinA=h/b and SinC=h/c. if we get these two formulas and divide them with each other we notice that we end up with the law of sines being SinA/a=SinC/c. 
Area formulas - How is the “area of an oblique” triangle derived?
The formula of an oblique triangle that everyone know is Area=(h*b)/2. What we do here is that we want to use this same formula but to find a missing angle on the triangle that we don't know. Since we dont care about finding h what we have to do is substitute it. Through the knowledge that we have have of trig functions we know that we can replace h with something else. In this case  h would equal a times the Sin A. this would be plugged in inside our old formula and eventually we would reach the point where our new formula would equal a = 1/2b(csinA)

This video went through how the formula is used and gives a more clear example oh how the formula comes from and how it should be used for these kinds of triangles. 
Law of sines picture

WPP 13/14 Unit P concepts 6/7

This blog post has been made in collaboration with Damian G and Tommy O. Please click HERE to see the WPP along with other very cool posts. 

Wednesday, March 5, 2014

WPP #12: Unit O Concept 10-Angle of Elevation and Depression

a. Marty is standing on top of the stage and he's about to start his show. He spots his mom in the audience and wants to know how far away is she. The angle of depression is 38 degrees and the height of the stage is 28 feet. How far away is Marty's mom?
b. One of the fans wants to know how far away she is from the stage. The height of the stage is still 28 feet. The angle of elevation is 5 degrees. How far away is the fan?