Week 05 · Tangent, Symmetry and the Turn
Dock Day
Dock Day: a shadow at home
// The mast's shadow
⏱ about 20 min
Dock Day: A Shadow at Home
Saturday the crew sends everyone home with chalk, a tape and one stick.
"Measure the stick and its shadow," Comet says. "Divide. Find the nearest special tangent."
"Then do it again later," Wren adds. "The ratio changes, and so does the angle. Show the family why."
Nova dims to a soft glow. "Would you like a hint? Draw the triangle before you divide."
"And one more thing," Comet says. "Draw the unit circle and show them 30° and its three reflections."
"Four points, one rectangle," Wren says. "The fold lines are the proof."
- Tangent is sine over cosine: opposite over adjacent, rise over run, height over shadow.
- The two special triangles give tan 45° = 1, tan 30° = √3/3 and tan 60° = √3. Sine and cosine sit beside them.
- π - x keeps sine, π + x keeps tangent, 2π - x keeps cosine. The reference angle gives the size, the quadrant gives the sign.
- Cosine is even, sine and tangent are odd. Sine and cosine repeat every 2π; tangent repeats every π.
◇ FAMILY SHADOW, THREE TIMES
Outside on a sidewalk, or indoors with a lamp, stand a stick upright and chalk the tip of its shadow.
Measure stick and shadow. Divide. Compare with 0.58, 1 and 1.73 and name the nearest special angle.
Repeat later, or move the lamp. Someone else predicts whether the angle went up or down before you divide.
Draw a unit circle together. Mark 30°, 150°, 210° and 330°. Fold along each axis to show the matching points.
Measure stick and shadow. Divide. Compare with 0.58, 1 and 1.73 and name the nearest special angle.
Repeat later, or move the lamp. Someone else predicts whether the angle went up or down before you divide.
Draw a unit circle together. Mark 30°, 150°, 210° and 330°. Fold along each axis to show the matching points.
For a grown-up
This is a measuring activity with a stick, a tape and chalk. Never look at the Sun or straight into a lamp.
The stick stays on the ground or in a grown-up's hands. Nothing is climbed.
A ratio that lands between two special angles is a good moment: the real angle is in between, and that is fine.
The stick stays on the ground or in a grown-up's hands. Nothing is climbed.
A ratio that lands between two special angles is a good moment: the real angle is in between, and that is fine.
DOCK DAY CHECK
- Read the question.
- Tap your answer.
A stick 1 m tall casts a shadow 1 m long. What is the light's angle, to 1 decimal place?
The shadow is about 1.73 times the stick. Which exact tangent matches stick ÷ shadow ≈ 0.58?
On your unit circle, what is sin 150°?A stick's shadow is three times as long as the stick. Is the light's angle above or below 45°? Type one word.
Stick ÷ shadow is 1/3, which is less than tan 45° = 1.
WHY THIS EXERCISEA smaller tangent means a smaller angle. Long shadows mean low light.
Draw your home stick and its two shadows as two right triangles, with the angle you found written in each.