Unit Q Concept 4

Multi-Level Problems:

Unit Q: Blogpost for concepts 2-3

tanx = rad3 / 3; cosx = -rad3 / 2
Special right triangles apply here.
Understand that:
tan = y/x
cos = x/r

when tan = rad3 / 3, the angle is 30 degrees.
when cos = - rad3 / 2, the angle is 30 degrees.



Now that we understand this, we have to find the value of the ratio when the angle is 30 degrees, so we must see that the point on the unit circle will be (rad3/2,1/2). We use this for all of the ratios.

Understand this next:
sin=y/r=1/2
csc=r/y=2
sec=r/x=2rad3/3
cot=x/y=rad3


Now that we are this far, we must remember that the angle could be in multiple quadrants. Tangent is positive so we automatically know it could be in the first or 3rd quadrant. Cosine is negative, it could be in the 2nd or 3rd quadrant. They both have the 3rd quadrant in common, so it has to be there.

sin = -1/2
csc = -2
sec = - 2rad3 / 3
cot = rad3

Using identities to solve the same problem.

We must understand all this:
sin^2 = 1 - cos^2
sin^2= 1 - (-rad3/2)^2
sin^2 = 1 - 3/4
sin^2= 1/4
rad sin^2 = rad1/4
sin = 1/2 or -1/2

csc = 1/sin
csc = 1/ (1/2)
csc = 2

sec = 1/cos
sec = 1/(-rad3/2)
sec = -2 / rad3
AT THIS STEP WE RATIONALIZE
sec = -2rad3 / 3

cot = 1/tan
cot = 1/√3/3
cot = 3/√3
RATIONALIZE THE DENOMINATOR
cot = 3rad3/3
cot = rad3

Deriving Pythagorean Identities

First, we set up a right triangle inside the unit circle. We label the sides appropriately. In a triangle, we know that sine = y/r and cosine = x/r
X is the adjacent side of the triangle, while Y is usually the opposite side, and the R equals 1.
With this information, we can substitute in values, where sin = y/1 = y and cos = x/1 = x

Now thinking back to the Pythagorean Theorem, we know that it says if you square the sides of the triangle, add them, we end up with the hypotenuse. In other words (or variables, I should say), x^2+y^2=r^2

NOW REPLACE THE VARIABLES!

cos^2+sin^2 = 1


It has now become one of the three Pythagorean identities, the other two can be easily derived from this one. Let's have a look now.

The next identity is 1+tan^2theta=sec^2theta

So how do we get this? Well, take the first formula and divide it by cosine. The first part should look something like sin^2/cos^2 (THIS IS Y/X, AKA, TANGENT). The second part is cos over cos, which is equal to 1. The last part is 1/cos (THIS IS SECANT!)
So now we simplify it and get:
1+tan^2theta=sec^2theta

Pretty simple right?
Now I will show you how to derive the third identity!

Goal to make
1+tan^2theta=sec^2theta
look like
1+cot^2theta=csc^2theta





This is accomplished by going back to the first identity and instead of dividing by cosine, we will divide by sine.

Following the same steps as before, we should end up with sine over sine (1) and cosine over sine (RECOGNIZE THIS AS COTANGENT) and 1 over sine (RECOGNIZE THIS AS COSECANT)
Now we simplify, and end up with:

1+tan^2theta=sec^2theta

Reflective Blog post

1. How have you performed on the Unit O and P tests?  What evidence do you have from your work in the unit that supports your test grade (good or bad)?  Be specific and include a minimum of three pieces of evidence.

I received a C+ on the unit O test, and retook it. I had done all my work correctly the first time, but the reason I received a C is because of small mistakes. Unit P I did bad on because I did not get enough help with law of cosine.

2. You are able to learn material in a variety of ways in Math Analysis.  It generally follows this pattern:

→ Your initial source of information is generally the video lessons and SSS packets followed by a processing and reflection activity via the WSQ
→ individual supplemental research online or in the textbook before class
→ reviewing and accessing supplementary resources provided by Mrs. Kirch on the blog
→  discussion with classmates about key concepts
→ practice of math concepts through PQs
→ formatively assessing your progress through concept quizzes
→ cumulatively reviewing material through PTs
→ Final Assessment via Unit Test.

Talk through each of the steps given in the following terms:
a. How seriously do you take this step for your learning?  What evidence do you have to support your claim?  Make sure to make reference to all 8 steps.
step 1: I use the SSS for everything, I always keep it with me and make sure I have highlighted what is important.
step 2: I don't use the text book at all, and only videos if I need help on something.
step3: I don't use resources on the blog at all.
step4: I depend on the PQ's to give me practice and allow me to understand the material, this is something that is important to me.
step5: Discussion is very important to me too because I learn and ask questions what I am confused on.
step6: Quizzes let me see what else I need to study, they are important to me.
Step7:The PT is important to me too becuase it's basically one of the last study tools that I use right before the test.
Step 8: The test itself is important because it's worth a lot and can make or break me.

b. How could you improve your focus and attention on this step to improve your mastery of the material?  What specific next steps would this entail?  Make sure to make reference to all 8 steps.


step1: I could avoid just doing what is needed and fill in all the boxes or problems with work. I usually just do what is needed.
step2: I think I am doing the most i can with this step, the videos i use and learn from very well now.
step3: I could use the extra resources now because sometimes the videos are not clear, I suppose this is why those resources are there, to clarify things.

step4: I believe I am doing everything correctly and efficiently with this step, the PQs are important to me.
step5: I believe that the discussion that happens with my group is also done very well. I learn things from others that I may have been confused on.
Step6:The quizzes could be retaken for full credit, sometimes I do not do this. I could retake those which do not get high scores on.
step7: I believe I do the PT to the best of my ability. I always spend a lot of time on it.
Step 8: I should review the material prior to the test from quizzes which I know i struggled on.



3. Reflect on your learning this year thus far by considering the following questions:
a.  How confident do you generally feel on the day of a Unit Test?  Give evidence and specifics to back up your answer.I usually feel nervous, but confident at the same time because I study the night before. Anything that I'm still confused about I basically ask other students or go at lunch to watch a video on it.
b.  How well do you feel you have learned the math material this year as compared to your previous years in math? Give evidence to support your claim.

I feel like I haven't learned as well as last year or the year before. I never struggled this much in the algebra 2 honors course. I think it's only specific areas where I have trouble in.
c.  How DEEPLY do you feel you have learned the math material this year as compared to your previous years in math?  Give evidence to support your claim.

I definitely have learned the material with a greater depth. Basically, I know how or why something is in mathematics now. It's not just a mindless following of steps. Just like we derive formulas, we never did that in other years.
d.  Do you normally feel like you understand the WHY behind the math and not just the WHAT/HOW?  Meaning, do you understand why things work, how they are connected to each other, etc, and not just the procedures?  Explain your answer in detail and cite specific evidence from this year.
I normally do understand this, just sometimes it's hard to follow when someone explains it. I usually learn by learning the "what and how" part first, and then learning the why. Just like when we derived formulas (law of sine/cosine), I first learned how to solve a problem using the formula, and then i learned to derive it.

e. How does your work ethic relate to your performance and success?  What is the value of work ethic in real life?

I have bad work ethic. I have a bad grade. Work ethic shows the ambition someone has for something.

WPP 14

Create your own Playlist on MentorMob!

WPP 13

Create your own Playlist on MentorMob!