Week 06 · The Addition Formulas
Dock Day
Dock Day: two rulers at home
// Two oars at an angle
⏱ about 20 min
Dock Day: Two Rulers at Home
Saturday the crew sends everyone home with two rulers, a protractor and a sheet of graph paper.
"Hinge the rulers with your thumb," Comet says. "Set 45° and 30° more. Measure the height of the tip."
"Then ask the family what sin 75° should be," Wren adds. "Most people will add. Show them why that fails."
Nova dims to a soft glow. "Would you like a hint? Start with the size check. Sine never passes 1."
"Then the formula, then the ruler," Comet says. "Three answers that agree."
"And one bonus," Wren says. "Set both rulers to the same angle and show sin 2a = 2 sin a cos a."
- sin(a + b) = sin a cos b + cos a sin b. And cos(a + b) = cos a cos b - sin a sin b. Angles add; sines and cosines mix.
- Replace b with -b, using sin(-b) = -sin b and cos(-b) = cos b, to get the subtraction formulas.
- tan(a + b) = (tan a + tan b)/(1 - tan a tan b), by dividing sine by cosine.
- The formulas give exact values for 15°, 75° and 105°: sin 75° = (√2 + √6)/4, sin 15° = (-√2 + √6)/4.
- Put a = b for the double angles. Then sin 2a = 2 sin a cos a and cos 2a = cos²a - sin²a.
◇ TWO RULERS, THREE ANSWERS
Hinge two rulers at one end. Set 45° with the protractor, then 30° more. Ask everyone to guess sin 75°.
Someone adds the two sines and gets about 1.21. Someone else checks: can a sine pass 1?
Compute sin 75° by the formula and as a decimal. Then measure: a 10 cm mark on the top ruler, and its height.
Divide the height by 10 and compare. Trade roles and try 15°, then both rulers at 30° for the double angle.
Someone adds the two sines and gets about 1.21. Someone else checks: can a sine pass 1?
Compute sin 75° by the formula and as a decimal. Then measure: a 10 cm mark on the top ruler, and its height.
Divide the height by 10 and compare. Trade roles and try 15°, then both rulers at 30° for the double angle.
For a grown-up
This is a measuring activity with rulers, a protractor and graph paper. Rulers are held flat on a table.
A guess that adds the sines is the best teaching moment of the day: the size check shows it cannot be right.
The angles and heights are the family's own readings, not facts about any real oar.
A guess that adds the sines is the best teaching moment of the day: the size check shows it cannot be right.
The angles and heights are the family's own readings, not facts about any real oar.
DOCK DAY CHECK
- Read the question.
- Tap your answer.
Both rulers at 45°, so the top one sits at 90°. Using sin(a + b), what is sin 90°?
A bonus: cos(90° - 30°). Use the subtraction formula. What is it exactly?
The 10 cm mark on the top ruler sits at 75°. How high is it above the table? Round to 1 decimal place.Is sin 75° bigger or smaller than sin 45° + sin 30°? Type one word.
sin 75° ≈ 0.97 and the sum is about 1.21.
WHY THIS EXERCISEThe second oar tilts, so part of its length leans sideways instead of rising. The sum is always too big.
Draw your two hinged rulers with the angles marked and the height you measured at the 10 cm mark.