Mathochism: Piecewise frustrations

One woman’s attempt to revisit the math that plagued her in school. But can determination make up for 25 years of math neglect?

Under the compass of Damocles We had our first quiz today. It was a precalculus review. We were supposed to have it last Thursday, but the professor was behind on the material, so we got a reprieve and a chance to really study over the long weekend.

I used that time to bone up on my inequalities, my trig identities and root functions. I spent a total of 12 hours or so, fueled by a pan of brownies, madeleines and tea, poring over various books, including the excellently comprehensive summer pre-calc book.

I didn’t get a great night’s sleep last night, though, partly because I sleep poorly when I’m nervous, partly because there was a problem in the professor’s trig review I just couldn’t solve, and partly because my spouse is sick, or allergic, or something, and snored and wheezed all night. Normally, I deal with his snoring by shoving him somewhat gently, but he had been so miserable all yesterday that I didn’t have the heart. Also, payback is a bitch.

Still, I did get some sleep, and spent the morning just going over stuff I just needed to memorize. And I got a bottle of Peach Snapple Ice Tea, because that has become a ritual on test days.

So, the quiz. Here are the positives:
1) It was completely fair in testing material we had actually gone over, but not ridiculously simple.
2) I aced the trig part, which I had been agonizing over (see problem I couldn’t solve, above).
3) I was able to apply math test anxiety techniques I learned earlier this year, including the brain dump and staying away from nervous classmates (though I was tempted to smack one loud fretter in the hall).
4) It is a quiz, not a test, so tanking this is not a tragedy.

The negatives:
1) I have some sort of block about piecewise equations, which makes me forget simple concepts like y = 4 is a horizontal line.
2) I over-thought the f of g problem. I set it up right, but simplified it too much, multiplying things out that were best left alone, and in the process, forgot to identify zeroes in the domain. Urrgh.
3) I got the transformation on a root function graph right, but skipped a step. My answer is correct, but the professor will likely take off points.
4) I misplaced a negative sign in a slope equation, which is infuriating.

Since we went over the quiz after we took it, I know what mistakes I made. Most can be attributed to nervousness, not to a lack of understanding of the subject. This is a good thing. Last weekend, when I was testing myself on all the concepts, I was gratified to see I scored solidly in the 90th percentile. I just need to plug on.

So far, this professor is giving me a much better vibe. I hope it lasts!

PS: Can any kind mathematician out there help with this problem? I actually went to a tutor before the test, but a lot of students were in line before me, and he didn’t have time to help me before class. Fortunately, that example was not on the quiz!

Verify 2 sin x²(π/4 – x/2) = 1 – sin x

2(sin π/4 – sin x/2)² = 1 – sin x ( Working just with the left side, I rearranged it, recognizing it as an addition identity)

2(√2/2 cos x/2 – √2/2 sin x/2)² = 1 – sinx (I applied the addition identity, and turned sin π/4 and cos π/4 into √2/2.)

(√2 cos x – √2 sin x) (√2cos x – √2 sin x) = 1 – sin x (I got rid of the 2 at the front.)

2 cos² x – 4 cos x sin x + 2 sin² x = 1 – sin x (I multiplied out the expression.)

cos² x – 2 cos x sin x + sin² x = 1 – sin x (I divided by two.)

1 – 2 cos x sin x = 1 – sin x (I recognized the Pythagorean identity, and simplified.)

1 – sin 2x = 1 – sin x (I recognized the double angle sine identity, and simplified.)

1 – sin 2x does not equal 1 – sin x!!!

Help, kind mathematicians! What did I miss?

All text copyrighted by A.K. Whitney, and cannot be used without permission.

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2 comments

  • In going from
    “2(sqrt2/2cosx/2 -sqrt2/2sinx/2)^2 = 1 – sinx (I applied the addition identity, and turned sin p1/4 and cos pi/4 into sqrt2/2.)”
    to
    “(sqrt2cosx – sqrt2sinx) (sqrt2cosx – sqrt2sinx) = 1 – sinx (I got rid of the 2 at the front.)”
    you got rid of too many twos.

    The second line here should be
    (sqrt(2) cos(x/2) – sqrt(2) sin(x/2)) *(sqrt(2)cos(x/2) – sqrt(2)sin(x/2) = 1 – sin(x).

    Otherwise, it looks fine.

  • Thank you! I’ll try again.