BQ#2 – Unit T Concept Intro: How do the trig graphs relate to the Unit Circle?

Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
The reason why sine and cosine have a period of 2pi is because that's how long it takes for a sine/cosine graph to repeat itself. The repetitions are based on the signs of the trig function and the unit circle. Whereas tangent, tangent repeats the sign pattern in the unit circle, so for it to complete a repeated cycle. So in 2pi there would be 2 graphs looking exactly the same for that reason only 1 pi is needed for the graph. 

 
Amplitude? – How does the fact that sine and cosine have amplitudes of one (and the other trig functions don’t have amplitudes) relate to what we know about the Unit Circle?
We have to remember that in the unit circle, a value greater than 1 and less than -1 would result as an error, because the values cannot be greater than that on the unit circle. for that reason those are used as amplitudes while graphing a sine/cosine graph. 


Works Cited
http://edunettech.blogspot.com/2013/10/the-graphs-of-six-trigonometric.html 

Reflection# 1: Unit Q, Trigonometric Identities

What does it actually mean to verify a trig identity?

When we are asked to verify an identity what we have to do is make sure that both sides equal each other exactly. we can do this by substituting trig functions with each other. No matter what, both sides equal the same thing. we just have to check that its true. 

What tips and tricks have you found helpful? 

Well the best tip for this unit is memorize the identities. If you know your identities you can look at a problem and be ahead of the problem. There were times where i knew what i was going to do in the problem two steps before i actually did them. This was thanks to me memorizing the identities. 

Explain your thought process and steps you take in verifying a trig identity.  Do not use a specific example, but speak in general terms of what you would do no matter what they give you. 

The first thing that i would see is what are the terms in which we have to end up with. Is it going to be a fraction? Do they cancel at the end? these are some questions that go through my mind in the beginning. then if i have a fraction i try to get rid of the bottom part if possible. then i try to set everything to sin and cosine. so that later i can just move those two around to turn them into the trig function it has to equal to.  

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. 
WORKS CITED
Law of sines picture
http://www.drcruzan.com/Images/TrigNonRight/LOS_Derivation.png 
 

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. 

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?

I/D# 1: Unit N Concept 7: Special Right Triangles and the Unit Circle

INQUIRY ACTIVITY SUMMARY

As we know there are several right triangles. However, there are three examples of right triangles that we call "Special Right Triangles." These triangles are the 30, 45, and 60 degrees triangles, which have some special features as shown in the pictures below. 

http://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/6d31e6a7-f698-4f44-b2d2-953443b2e5bd.png *(edited by me)

30 degrees

First we labeled the rules of this special right triangle. Them being the hypotenuse is 2x, vertical value = x and horizontal value = x√3. 

The next step to this triangle was to find a way to which we can make the hypotenuse equal 1. The way to do this is to divide 2x by itself making it equal 1. As we know, whenever a change is done to a side, the same change must be done to the other sides. Due to this, we divide x and y by 2x leaving us with horizontal value = √3 / 2 and vertical value = 1 / 2

Now you just equal the hypotenuse to r, horizontal value to x and vertical value to y. r = 1, x = √3 / 2, and y = 1 / 2.

Then you draw a plane on the triangle so that the triangle lies on quadrant I

Finally you label the three points on the triangle. The points for this triangle should be (0,0), (√3 / 2, 0), (√3 / 2, 1 / 2)

45 Degrees

First we labeled the rules of this special right triangle. Them being the hypotenuse is x√2, vertical value being x and horizontal value also being x.

The next step to this triangle is to find a way to which we can make the hypotenuse equal 1. The way to do this is to divide x√2 by itself making it equal 1. As we know, whenever a change is done to a side, the same change must be done to the other sides. Due to this, we divide the horizontal and vertical values by x√2 leaving us with horizontal value = √2 / 2 and vertical value = √2 / 2.

Now you just equal the hypotenuse to r, horizontal value to x and vertical value to y. r = 1, x = √2 / 2, and y = √2 / 2. 

Then you draw a plane on the triangle so that the triangle lies on quadrant I

In the end, you label the three points on the triangle. The points for this triangle should be (0,0), (√2 / 2, 0), (√2 / 2, √2 / 2)

60 Degree

First we labeled the rules of this special right triangle. Them being the hypotenuse is 2x, vertical value = x√3 and horizontal value = x.

The next step to this triangle was to find a way to which we can make the hypotenuse equal 1. The way to do this is to divide 2x by itself making it equal 1. As we know, whenever a change is done to a side, the same change must be done to the other sides. Due to this, we divide x and y by 2x leaving us with horizontal value = 1 / 2 and vertical value = √3 / 2.

Now you just equal the hypotenuse to r, horizontal value to x and vertical value to y. r = 1, x = 1 / 2, and y =√3 / 2.

Then you draw a plane on the triangle so that the triangle lies on quadrant I.

Finally you label the three points on the triangle. The points for this triangle should be (0,0), (1 / 2, 0), (1 / 2, √3 / 2)

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

This activity helped me not only to find the points that we will be using in the unit circle later in the unit but also to visualize where those points came from. Now these points are not just random numbers that were thrown at me to memorize. I understand the background of the numbers, and how they relate to the angles. This activity will also help when I have to use the trig functions because I will know which numbers I have to use for each one of the functions. In conclusion, this activity helped me understand where the unit circle's values come from and how I can use them later in the unit.

Quadrants and Signs

Something that is very interesting about these triangles is that once you found the I quadrant you can figure out what the other sides are because the values will be the same the values will just change signs. First,you base yourself on the reference angle to measure the 30, 45, and 60 degrees.Once you do this you can start assigning the numbers. On Quadrant II , the numbers will be the same. The only thing that will change will be the sign on the x value. On Quadrant III, you still base yourself on the reference angles, then the change on this quadrant will be that both the x and y value will be negative. Finally, in Quadrant IV they only change that has to be done is that the y value will be the only one negative. x will stay positive. 

INQUIRY ACTIVITY REFLECTION

The coolest thing I learned from this activity was…how everything that we learned in the unit falls together through something like this (unit circle). I also found very interesting how past courses such as geometry and algebra II had such a great influence on something like this where you relate angles with circles and triangles. 

This activity will help me in this unit because… it will help me not only know the values that i will be needing when I need to use the unit circle but to understand where it comes from. So that in case I ever forget how all these numbers came to be and I forget the values I know where I can get these values

Something I never realized before about special right triangles and the unit circle is… how the both of them compliment each other so much. I never thought that through triangles I'd understand how a circle works and through that we can find different values along with the trig functions.