precalc 21 2 precalc 21 3 precalc 21

 

What do these responses reveal about what the students know about trig? Or just comment about what you find interesting here.

Thanks to the Incredible Tina Cardone for the submissions.

  • It really looks like these students don’t know the Sine identities very well (or at all?). I’d recommend students try this one out on their calculator to see that their results don’t work, as logical as they seem. One would then need to spend some more time with the sine identities. I wonder how you could explore to find the sine identities in a remotely efficient manner…

  • I think these students don’t understand the difference between a “function” and “regular old parenthesis”. I wonder if they think this is true:

    f(75) = f(30+45) = f(30) + f(45)

    If only we had used different notation for “functions” and “multiplication” however many hundred years ago this was established… but students have to learn the silly little notation differences that we take for granted. (Don’t get me started on sin^-1(x)…)

  • This problem jogged my memory to supply this, um, off topic response, but 75° angles play a part in this problem as well.

    http://fivetriangles.blogspot.com/2013/02/45-overlapping-circles.html

  • Looks like students are clear on linear functions (and perhaps functions & function notation in general). If I’m recalling the traditional pre-calc course correctly, this immediately precedes solving things like sin(x + 30) = [sqrt(2) + sqrt(6)]/4. Would introducing (without first introducing inverse trig functions) help at all? Kind of how you might ask a child “What plus 2 is 6?” before formally introducing subtraction.
    /notamathteacher

    • meant to say “NOT clear on linear functions.” smh

  • The second and third answers are greater than 1. The second is also far too small to be sin(75). (For instance, the student knows sine of 60. If she also know that sine increases from 0 to pi/2, she might realize that answer should be bigger than sine of 60, not smaller).

    If there were time to have students explore the geometric properties of multiplication of complex numbers, it leads to a nice exploration of the cosine and sine addition formulas. Even if there’s not time for that, one could memorize the identity

    cis(theta1) x cis(theta2) = cis(theta1+theta2)

    as a mnemonic tool for remembering the addition formulas. Expand the left side and equate real and imaginary parts to get both formulas.

    • Oops, I meant the first is far too small to be sin(75).

  • This was given as part of the no calculator portion of the test, and students had a trig identities sheet available which included the rule for sin(a+b). Getting across the message that functions don’t work like the distributive property is an uphill battle, especially since function notation is used so infrequently. Things I wish happened in Algebra 1…

    • Haha, give me a nickel for every fact I wish stuck from Algebra 1… beggars would ride (or something like that).