davidwr's Journal: Create 2-digit trig table by memorizing 6 numbers

Jr. High/High School math teachers:
Want to teach your students to be able to generate a trig table with a 4-function-plus-square-root calculator?

Have them memorize:
sin 0=0 (exactly)
sin 10=0.174
sin 20=0.342
sin 30=0.5 (exactly)
sin 37=0.602
sin 45=0.707 (sqrt(2) exactly)

From these, they can interpolate the rest of the sin table from 0 to 45 degrees.

They can generate a cosine table for 0-45 degrees using the formula:
1 = cos^2 + sin^2
or
cos = sqrt(1 - sin^2)

For 46-90 degrees use the formula:
sin(x) = cos(90-x)
and for 0-90 degrees which is what we care about:
cos(x) = sin(90-x)

Tangents are simply sin/cos.

Once you know these, cotangents, and the inverse functions fall into place.

The resulting table is good to within 0.5%, or 2 significant digits.

As a science-tie-in bonus, sin37 is approximately 0.602, which "looks like" Avogadro's number, 6.02 x 10^23, the number of atoms or molecule in a mol of a substance.

Additional observations for students:
1) interpolations are frequently good enough
2) the sin curve is relatively flat in 0-30 degrees so we can have our interpolation points spaces further apart. Using 10,20,30, 40, and 45 as interpolation points leads to inaccuracies greater than 0.5% between 30 and 40.

For advanced students:
What two points between 0 and 30 generate the table with the best worst-case error for 0-30 and 60-90 degrees for sin? for cosine? for tangent? [answer: I didn't calculate this yet.]
Why were 10 and 20 chosen? [answer: it's easy to divide by 10 which makes interpolation easy.]

What point between 30 and 45 generates the best worst-case error for 30-60 degees for sin? for cosine? for tangent? [hint: 37 is better than 38].