Precal Blog

I hope you all really enjoy your time off – you have certainly earned it. If you post or comment during break you may not see it show up right away. Don’t worry – it will. Don’t post or comment twice, just be patient. -Peace

Today we started by going over the worksheet we did with the sub yesterday. We found out that the inverses we’d graphed were relations, not functions, and they are called arcsine, arctangent, and so on. By restricting the domains on these graphs we can produce functions — the area that is counted as the actual inverse of the function is called the principal branch.

We found out several criteria for selecting the principal branch:

1. The selection must be a function.
2. It must use the entire domain of the inverse relation. (And it was noted that the domain and range of a function and its inverse are reversed)
3. It should be continuous if possible (sometimes it isn’t).
4. It should be centrally located (near the origin).
5. Use the positive branch if there are 2 choices (see cot and cos).

We then discussed the range and domain of the inverse functions, as so expertly graphed by Kari.

Finally, we got a sort of quick example of the kind of problems we will be given with this information.

For example, `cos(tan^(-1) (4/3))=(3/5)`

First, we’d look at the tangent, which is opposite over adjacent:

And then we’d consider the cosine:

And also notable:

Enjoy tomorrow’s party as reported here by KatelynW and allow yourself a sigh of relief because winter break has arrived

Shelby was picked to scribe today, but apparently she has scribed 3 times. So Kaitlyn picked me to scribe today instead.

Today in class we discussed the worksheet we did yesterday in class. It was the worksheet about the inverse of sin, cos, tan, cot, sec, and csc. These are also called arcsin, arctan, and so on.

We first discussed arcsine

The graph looks like:

This graph was created by reversing the x and y coordinates in the original sine .

The reason only a portion of the graph is shown is because the whole relationship is not considered a function because it fails the vertical line test.

To find this portion there are some rules to follow:
1. It should contain the entire domain of the function.
2. It should be as close to 0 as possible.
3. It shoud be positive if possible.

Once we determined which part to focus on, then we had to figure out the domain and range of this ‘function’. Remember to write in the correct notation. Mr. B is going to be picky about that now.

The Domain of arcsin is [-1,1]. (brackets mean that number is included. Regular parentheses means that the number is excluded.)

The Range of arcsine is `[pi/2,pi/2]`

We then tackled arccos which is similar to arcsin.

The domain is [-1,1].
The range is `[0,pi]` this time though.

Sec is `1/cos(x)` so arcsec will have similar range of arccos.

The domain is `(-infty,-1]U[1,infty)`.
The range is `[0,pi/2)U(pi/2,pi]`

Csc is `1/sin(x)` so the arccsc will have a similar range to arcsin:

The domain is `(-infty,-1]U[1,infty)`
The range is `[-pi/2,0)U(0,pi/2]`

arctan is another graph we looked at in class.

The domain this time is `(-infty, infty)`
The range is `(-pi/2,pi/2)`

The last trig graph we did in class is last but not least the arccot:

The domain of this graph is also `(-infty,infty)`.
The range of this graph is `(0,pi)`.

That was all we did in class today. Remember tomorrow we are having an Office Christmas Party. I think we are even ordering pizza fourth hour.

I will leave Amy the hard task of blogging over break.

I enjoyed reviewing the photographs you took for this assignment. If you can e-mail me the digital pics I’d like to put up a slide show on the blog so everyone can see your work. In fact Flickr has a neat tool which lets you annotate your photograph with click-able boxes that could, for example, reveal information about your sinusoid or its equation.

I thought Blair submitted a noteworthy entry. His friend has a snake that has been photographed by National Geographic. Cool!! You can see it here.

Today we came to class to find that Mr.B was not there and that we had a sub. But Mr. B did have work for us to do. He left us a worksheet on the graphs and inverse graphs of Sin, Cos, Tan, Csc, Sec and Cot. Producing the inverse graphs of Sin, Cos, and Tan graphs were easy, but the inverse graphs of Csc, Sec, and Cot were a bit more dificult to produce. To make the inverse graphs of Csc, Sec, and Cot, you have to put you calculator into parametric mode. Then in equation editor(y=) for X put Sin(T) or Cos(T) ect.ect. For the Y put “T” no quotes. Our homework for tonight was to do some more problems from Section 4-3.
Tomorrows Scribe will be Jessica.

A few days ago I was conversing with my good friend Laura. As we were talking, the topic of her birthday came up. “What would you like for your birthday, dear friend Laura?” I asked. She replied, “Why Kaitlyn I would like nothing more than to learn about inverse trigonometric functions!” Believe it or not Laura’s birthday wish came true! What a lucky gal!

Today we filled out a worksheet of parent circular functions on coordinate planes. The first page was simple enough, we were just asked to graph sin, cos, tan, cot, sec, and csc in their normal form. The coordinates were dealing with radians rather than degrees so where the sin graph would have touched 180 degrees it met the pi mark. Just as a reminder: 90 degrees is the same as pi over two, pi is the same as 180 degrees, 270 degrees is the same as three pi over two, and two pi is the same as 360 degrees. Your graphs should look like the trig function graphs we’ve been doing for the last few units.

We were then asked to graph the inverse relation for each of the six trig functions. This was done simply by switching the x and y coordinates. Many students were confused because the believed that sec and csc would be the inverse graph for sin and cos but, these these relations represent the reciprocal (or one over) sin and cos- not the inverse. When the x and y coordinates were switched the graphs appeared to have flipped from horizontal to vertical graphs. I found it beneficial to check the inverse graphs I had drawn on my graphing calculator. It helped in viewing the reversing of the x and y points. This part of the worksheet was summed up by stating: The graphs you created in part II are inverse relations because they fail the vertical line test. We distinguish these inverse relations from the actual inverse functions by calling them arcsine, arccosine, arctangent, arccotangent, arcsecant, and arccosecant.

The third part and final of this worksheet talked about using our calculator to plot these functions. If you type in sin-1(x), cos-1(x), and tan-1(x) they are appropriately graphed on the screen. The other 3 functions however are not. By observing these functions closer we find that they look like pieces of the inverse relations we graphed. Because Mr.B was not present today why this occurred was not fully explained but we will discuss it and the other three functions tomorrow in class.

Overall it seemed like Laura had a really great birthday! She got her wish of studying inverse trigonometric functions along with the added bonus of homework in section 4-3. If she’s lucky, maybe I’ll get her a book of Precalculus integrals, derivatives, and infinitesimals for Christmas!

The next scribe will be Shelby.

Today in class started with 20 minutes to work on the claims that were assigned on Monday. I was sick on Monday, so I had to work feverishly to complete as many as I could to get claims, and ended with two. Laura has already done a nice job explaining how the problems work, and since I didn’t copy down how our class did them, I’m going to assume they were done the same way, so just look at Laura’s post for the specifics.
Hannah started us of by doing an expert job solving #6, the first claim, and no questions were asked, meaning the explanation was perfect.

K-Stack followed this powerhouse performance with one of his own on #17. When asked for his reactions regarding receiving the problem to present, K-Stack said he “saw it coming”, and prepared accordingly

In between K-Stack and Phil’s presentation Mr. ——–( his name can’t be mentioned in the blog, and I can’t remember what we were supposed to call him, so he’s Mr. —– so no one can Google his name and see his court record) made a City Slickers reference, that was finally understood by Adam after numerous hints. This astounding feat should at least be worthy of counting as a presentation, at least in this bloggers opinion.

Phil’s presentation of #23 was pretty awesome, I can’t remember the specifics because I was still astounded by Adam’s pop culture mastery, so I’m just assuming it was good because he taught be before class.

Leroy was chosen to present #32, and he was “Shocked and Appalled” that Mr.—— would pick him. Even under such pressure Leroy gave a dominate presentation, with it being questioned only once by Britney. Britney’s question was deftly handled by Leroy, and soon the class had a better understanding of whatever he was teaching us, I didn’t pay attention because I was too busy writing down my entry for tonight.

Natalie was selected to present #49, a problem she described as a “Sticky Situation”. During her explanation she utilized Tim Foil, and numerous other formulas we recently learned. Her postulations covered both boards, and would have been kept for posterity had the Smart board been working, alas it was broken and her expert technique in solving this problem is lost to the ages.

No one in our class claimed #54, because as Leroy put it, it was “unsolvable”, but as I misquoted and Mr.—– wrongly attributed to Bob the Builder, “Nothing is impossible – not if you can imagine it!”(Hubert Farnsworth). So I dare you to do #54, cause I’m not doing it for you, I have a paper to write.

Tomorrow’s scribe is Hans.

so after being named scribe last monday, i am finally back in class and writing my post a week later for our class…

Today we spent about half the class just going over the claims with the other people in our class. After that, it was time to present

Luke did #6. he started off by doing cos^2x cscx secx to cotx. then he simplified that by taking (cosx*cosx)/1 and then making 1/sinx and 1/cosx. This gave him a final solution of cosx/sinx = cot x.

Liza did #17. She started out with (tann + cotn)^2 and then making (sec^2)n + (csc^2)n. she then used the distributive property to get (tann + cotn)(tann + cotn)(tann + cotn). Using the ever popular Tim Foil method a final phrase of tann^2 + cotn(tann) + cotn^2. the cotn(tann) cancelled out so that Liza had tan + 2 + (cot^2)n. With a little bit of class participation we reached the next step to Pythag the equation to 1 + (tan^2)n + 1 + cotn^2. This was (sec^2)x + (csc^2)x giving us a final product of (tan x + cot)^2 = (sec^2)x + (csc^2)x.

Next was Juliana who did #23. She begun by taking (secA/sinA) – (sinA/cosA) = cotA. This was made into 1/(cosA*sinA) – (sinA/cosA)*(sinA/sinA). Then she did (1-sinA/(cosA*sinA)). This is actually equal to (cos^2A)/(cosA*sinA). Eventually you are trying to find the cotangent of this problem so you are left with the cos on the bottom canceling out so you are left with cosA/sinA.

Kaitlyn presented #32. She did (cos^2)*theta + (tan^2)theta (cos^2)theat = 1. This changes into (cos^2)theta + (tan^2)theta (cos^2)theta. THAT expands into cosTheta cosTheta tanTheta tanTheat cosTheta cos Theta. that’s really long. The cosines cancel out to give you sinTheta/cosTheta sinTheta/cosTheta. This gives you a final of (cos^2)theta + (sin^2)theta = 1.

Grant was last with #54. He took (1 +sinx)/(1-sinx) = 2(sec^2)x * 2(sec^2)x * tanx – 1. luckily this shortens down into 2/((cox^2)x) + ((2 – tanx)-1)/(cos^2) -1. But it then expands to 2/((cos^2)x) + (2(sinx))/((cos^2)/x) – 2. he got this by multiplying both sides by 1-sinx and getting rid of the fraction. After a little more work, he was left with 1+sinx = (2-2sin^2)x)/cosx. turns into 2 = 2-(2(sin^2(x)))/(cos^2)x or 2(cos^2(x)) = 2 – 2(sin^2(x) you add and dcancel on both sides by 2 to get a fina of (sin^2)x + (cos^2)x = 1. mr b noted a correction the multiplication property of equality.

class ended with a reccommendaation that we work on lots of puzzles and problems to sharpen out skills until we feel comfortable. Tomorrow will bring the adventure of inverse functions

i haven’t really been here for the last week, as you know, so i hope i explained everyones problems good enough and i know i’ll definitely have to practice (and learn for matter) a lot of this stuff.

scribe for tomorrow is kaitlyn perry!

So…the semester is coming to a close on the 24th of January. This makes your final exam around the 22nd. You will be tested on the learning targets you have not yet shown proficiency on. You know what those are because you have all of your tests.

Since you have been keeping up on your quarter journal, you will only need to write your reflection piece and print out your new and improved comments from the blog. (See this post)

Lastly then, you are working on your portfolio which will also be due on the day of the final. Please reread the requirements for the portfolio often and pay attention to the scoring rubric.

What mathematical theorem is represented by the following mosaic? Justify your answer.