Below is something that’s been unique to my classes (as far as I know).  If you’ve taught trigonometry, you’ve seen the unit circle with the points marked for the special angles.  I have another tool that I use, which we call the BFT (Big Freakin’ Table).  The columns are “deg”, “rad”, and the six trig functions.  There are 17 rows, all blank, for the special angles from 0 to 360 degrees.  We fill it in as we learn the special ratios from the 45-45-90 triangle and the 60-60-60 triangle.  Then we expand the knowledge of trig to the ray passing through (x,y).  But presenting all of the data in this form allows me to point out some patterns and relationships that I think are key for good for understanding trig:

  1. Relationship between degrees and radians – the students will know how to convert from one to another, but they’ll also be able to see that pi/6 = 30 deg and 7pi/6 = 210 deg.  That kind of proportionality reinforces the understanding, in my opinion.
  2. Cofunctions – in the first quadrant, sin and cos are the same numbers but reversed order.  Also, they get that sin increases but cos decreases in the first quadrant.  The order of the columns reinforces the cofunction relationship; they’re all grouped together.
  3. Symmetry – they can see that sin repeats the same numbers coming back down in the second quadrant as going up in the first.
  4. Reference angles – they can see how even the negative numbers are repeats of the positive, just following the same pattern as sin pushes into the third quadrant.
  5. Identities – we compute tan and cot from sin and cos, and sec and csc by reciprocating cos and sin.  Done 17 times or so, this helps stick the relationship.  In fact, by the time we get to csc 210 degrees, there are several ways of easily figuring it out from the data on the table, and all of them work.  The order of the columns reinforces the reciprocal relationship; the reciprocals are the middle two (tan and cot), the next two out (cos and sec), and then the next two out (sin and csc).

When we’re done, I stress to them that from the quadrant 1 sin values we can get the quadrant 1 cos values.  From there we can get the rest of the quadrant 1 trig values.  Knowing how the reference angles work and the positive/negative pattern, we can get the entire rest of the table.  So all we need are 5 little values to get the entire table.  And I discovered a few years ago there’s even a pattern there:

  • sin 0 deg = sqrt(0)/2
  • sin 30 deg = sqrt(1)/2
  • sin 45 deg = sqrt(2)/2
  • sin 60 deg = sqrt(3)/2
  • sin 90 deg = sqrt(4)/2

















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One Response to BFT

  1. Pingback: Round Up of Week Two of the Math Blogging Initiation « Continuous Everywhere but Differentiable Nowhere

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