This is correct, so I will not bother restating it :D. I will add that this is taught in calculus 2, and if you have not had calculus it probably won't make any sense. However, if you have had calculus but it's been a while, here's a blog which includes two step-by-step derivations: http://blogs.ubc.ca/...ansion-of-sinx/ (note that the next-previous links at the bottom lead to the expansions for e^x and cos x)

A calculator will find trigonometric values using the power series expansion. A Maclaurin series (special case of the Taylor series) is an infinite series which approximates a function. You can get any desired degree of accuracy by taking as many terms as desired. For example, sin x = x - x^3/3! + x^5/5! - x^7/7! + ... This series converges extremely rapidly, and the error is easy to estimate because the signs alternate. The error is less than the next missing term -- for example, if I cut off the series after 4 terms, the error will be less than x^9/9!. Your calculations are fine. Most numbers raised to the integral multiples of i will not be "nice" any more than cos 1 is nice. (Btw, for a nifty trick, you can check your calculations by googling 2^(2i). Google calculator will automatically find it for you.)

You probably wouldn't notice it unless you specifically did a test. To her, it's normal. I definitely agree with getting paperwork on file for disabilities. It's better to have it on file and not need it than to need it and not have it.

1) Euler's Formula says that e^(ix) = cos x + i sin x. (you can read about this, here http://en.wikipedia.org/wiki/Euler's_formula for example) If a is a real number, by properties of logarithms a = e^(ln a). Therefore, a^i = (e^(ln a))^i. Since the normal rules of powers still apply, this = e^(i ln a), and by applying Euler's formula this is cos (ln a) + i sin (ln a). 2) a^(bi) = (e^(ln a))^(bi) = e^(i*b(ln a)). Therefore applying Euler's formula again, we get cos (b(ln a)) + i sin (b(ln a)). 3) a^(c+bi) = a^c times a^(bi) by laws of exponents, so multiplying a^c through number 2, we get a^c(cos (b(ln a)) + i sin (b(ln a))). Now, this assumes that a is a real number. It gets a bit more complicated if it's not. If you want more information or if I was unclear, let me know. (throughout, I have used ^ as notation for (to the power of))

Beginning and Introductory Algebra are two names for Algebra I The CLEP college mathematics is essentially a math for liberal arts course and should follow high school algebra and geometry. I wouldn't recommend it unless the student's already picked a college and knows they'll get college credit for it. Many won't give credit for this because they want you to take THEIR version of the math for liberal arts course. It is not the same as BCM, which is before algebra 1.

Sorry for the poor pic quality, but I'm delighted -- this is her first daughter.

(not physics, but math) You are absolutely not being unreasonable. He is being unreasonable. This is a huge cause of errors in math courses and I cannot imagine it's any better in physics courses.

Keys to Algebra is not a full Algebra 1 curriculum (despite what Cathy Duffy says) and will not prepare most students for Algebra 2. It is a great lead-in to Algebra 1.

I think this will be the key. It is difficult for most adults to find, identify, and remedy their own weaknesses. It is even more so for teenagers. If this does not work, I would strongly look into hiring a tutor, whether online or face-to-face. Fractions are one of the absolutely essential topics of arithmetic that must be understood before any sort of advanced math.

This is another great recommendation.

I would second the recommendation to look into AOPS for your dd -- if nothing else, add in the discrete math (number theory, counting and probability) texts for her between algebra 1 and 2. These subjects are tremendously fun and interesting, but there simply isn't room for them in the standard curriculum. AOPS curriculum is also very full and enriched -- it would definitely slow her down without shortchanging her education.

I have to add this cartoon to the centrifugal force discussion: http://xkcd.com/123/

I am assuming that you mean -x + (3/4)x as that gives the answer you're supposed to get. Try thinking about it like combining like terms. We know that -x + 3x = (-1 + 3)x = 2x. Similarly, -x + (3/4)x = (-1 + 3/4)x = -1/4 x

I think that moving forward with concepts *while continuing to work on facts at his level* is a great idea. I had a terrible time with facts and did not have them all memorized until much later in the curriculum. I am so thankful that I was allowed to work on concepts until I found where I really needed the facts. I could always derive them, but I did not have them at my fingertips.

Lial's is an extremely standard intermediate algebra text and should lead well into any college algebra text, so if she liked Lial's and did well with it for Algebra 1, that seems like a good option for continuing.

I would pick the one that YOU felt she was learning from the best out of all that you've tried, go back to it, put her where she places and move forward at HER speed. I would also take her opinion into account, although I would not give her the sole responsibility. Give the chapter tests that come with that curriculum and worry less about the standardized tests. Quit changing -- after this many curricula, you're not going to find a miracle. I'm not sure how you felt she was doing with MUS, but part of not doing well on the standardized test may also have come from the seriously atypical course sequence. I would expect someone who's gone through zeta to be ready for most standard pre-algebra curricula, and someone who's done pre-algebra to be ready for a highly rigorous pre-algebra or a standard algebra curriculum, but switching before that can cause (as you found) serious issues. ETA: I also agree with Scarlett.

How old is she, where did she test, and where was she in MUS at the time?

I don't have any specific knowledge of this place, but attending a non-accredited university may cause issues if you decide to pursue teacher certification or an advanced degree. Think carefully about your plans first.

These are too true to be funny :/

They usually come before College Algebra. Introductory is first, then Intermediate, then College. Introductory is sometimes also called Beginning. After College Algebra, the course names are not uniform. Sometimes Trigonometry and sometimes Precalculus, but both of these are fairly common. Content-wise, a student who has passed AND UNDERSTOOD (this is important) high school Algebra I ought to place into Intermediate. A student who has passed and understood high school Algebra II ought to place into College Algebra or Precalculus, depending on the strength of their high school curriculum. Even if the placement test score indicates a higher placement, I would not recommend placing higher than that. However, if the placement test score indicates a lower placement, I would go with the college's recommendation and assume that the basics had not been thoroughly mastered.

I would be really reluctant to ignore the life sciences completely. Although some colleges don't care, many colleges expect *some* sort of life science class. One reasonable approach would be finding one which is more to his liking (maybe one that focuses heavily on chemistry, or maybe an environmental science course with a heavy physical science component), and another reasonable approach would be picking a git'r'done biology curriculum and doubling on science one year. Similarly, I would be rather reluctant to ignore American history AND government completely. However, I would consider it reasonable to pick a relatively 'light' but still high-school level curriculum and work through it rapidly. Another option could be to cover American history within the context of world history and call the second year 'American History in a Global Context' or something similar. Yet a third option for both, if he is genuinely not interested in either, is to either self-study for the AP exam or dual-enroll during senior year (biology one semester, am gov one semester) with the goal of skipping college gen-ed courses and focusing more thoroughly on his major then.

This is exactly the type of learner for whom Saxon works. And if it is working for you and your student is doing well, enjoying it (or maybe even just tolerating it), and able to do math problems outside of the Saxon context, I would not switch. On the other hand, if you have one who *cannot* remember without seeing the big picture and cannot absorb knowledge in bite-size pieces, but *must* see the whole picture before being able to work problems, Saxon is a terrible choice. If you have one who needs far fewer problems than Saxon provides in order to grasp the picture and be able to understand, it may be essentially busywork. The rest of my comments are at the top of the thread.

I know. I caught someone blatantly cheating in an exam, and of course it was the student I liked best. I felt quite bad about it, but what could I do when someone was using their phone to quickly google a question during an exam?

I would try to get into the honors track as long as he is able to make good grades there. Trig is usually a (proper) subset of pre-calculus.

Yes, doing all the work on scrap paper is a bad idea. Doing initial thoughts is fine on scrap paper (I do this, and then copy the work AND the answer neatly into my typed file), but work should be included when copied. Otherwise, the student is unable to use previous work to study from. Don't feel like a failure. Many college students can't get a handle on this, so if your son can get this under control while in high school, he will be ahead of the game. See what his teacher says and go from there.