Tuesday

Comet swings the second oar back the other way. "Now it is 30° short of the first. 45° minus 30° is 15°."

"Same formula," Wren says. "But put in -30° for b. What do you notice about sin(-30°)?"

"Sine is odd, from last week," Comet says. "sin(-30°) = -sin 30°. And cosine is even, so cos(-30°) stays."

"So the plus becomes a minus," Wren says, writing. "sin 15° = sin 45° cos 30° - cos 45° sin 30° = (-√2 + √6)/4."

Nova shows a calculator reading 0.26. "Would you like a hint? Check every exact answer against a decimal."

"(-√2 + √6)/4 is about 0.26," Comet says. "Small, like a 15° angle should be."

"Two ways," Wren says. "The formula gives the exact value, and the decimal says it is sensible."

Replace b with -b in each addition formula. Use sin(-b) = -sin b and cos(-b) = cos b from week 5.

sin(a - b) = sin a cos b - cos a sin b. The sign in the middle flips.

cos(a - b) = cos a cos b + sin a sin b. The minus becomes a plus.

So cos 15° = cos 45° cos 30° + sin 45° sin 30° = (√2 + √6)/4. And sin 15° = (-√2 + √6)/4.

A calculator gives sin 15° ≈ 0.26 and cos 15° ≈ 0.97. Now evaluate the exact answers: (√6 - √2)/4 ≈ 0.26. They agree.

The decimal never replaces the exact value. It catches a wrong sign or a swapped term before you move on.

Try the check on a wrong answer: sin 45° - sin 30° ≈ 0.21, not 0.26. The formula was needed.

Divide sin(a + b) by cos(a + b), then divide top and bottom by cos a cos b.

The result is tan(a + b) = (tan a + tan b)/(1 - tan a tan b).

For the oars, tan 75° = (tan 45° + tan 30°)/(1 - tan 45° tan 30°). That is (1 + √3/3)/(1 - √3/3), which simplifies to 2 + √3.

Replace b with -b for the difference: tan(a - b) = (tan a - tan b)/(1 + tan a tan b).

Careful work. Tomorrow is Boathouse Lab: the cardboard oars get measured, and the formulas get tested with a ruler.