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Algebra 2 9-12 / Week 09 / Garden Day
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Week 09 · Trigonometry on the Unit Circle

Garden Day

Garden Day: the family circle
// Every angle has a point
⏱ about 20 min

Garden Day: The Family Circle

Saturday the string and chalk come out to the driveway, and the family draws a circle one big stride across.

"Each card names an angle," Comet says. "Walk to its point on the rim before Wren finds the coordinates."

The first card reads 120°. Someone walks past the top mark and stops. "Left and up," they say. "Cos -1/2, sin √3/2."

"How did you know the size?" Wren asks. "It is the 60° triangle, flipped across the up-down line."

Nova hovers over the chalk, her light a soft hint. "Would you like a card past one full turn?"

"Four hundred and five degrees," Comet reads. "That is a lap and 45 more. The 45° point."

"Next card," Wren says. "This one gives a sine and a quadrant, and asks for the cosine."

  1. A radian is an arc one radius long. A full turn is 2π radians, so 180° is π radians.
  2. Every angle has a point on the unit circle, (cos θ, sin θ). Cosine is x, sine is y.
  3. The special triangles give the first-quadrant values. The reference angle gives the size, the quadrant gives the sign.
  4. Angles that differ by a full turn share a point. Negative angles turn clockwise.
  5. sin²θ + cos²θ = 1, because the point sits on a circle of radius 1. Use it to find one value from the other.
◇ FAMILY CIRCLE CARDS
Chalk a circle with a string one stride long, and chalk two axes through its center.
Write eight angle cards: 30°, 45°, 120°, 180°, 210°, 300°, 405° and -90°. Shuffle them.
Draw a card and walk to its point on the rim. Someone else names the exact coordinates.
Check each with the string: lay it from the point straight to the x-axis and compare with the sine.
Trade roles so everyone walks a card and everyone names a point.
For a grown-up
This is a chalk and string activity on a dry, flat surface away from traffic. Pick the string up afterwards.
A wrong point is a good moment: ask which quadrant the angle lands in and check the signs.
The circle is the family's own drawing, not a measurement of anything real.
GARDEN DAY CHECK
  • Read the question.
  • Tap your answer.
A unit circle with a 120° angle marked from the positive x-axis; the point on the circle has coordinates (cos 120°, sin 120°).A card reads 120°. What is the sine, the y-coordinate of its point?
A unit circle with a 45° angle marked from the positive x-axis; the point on the circle has coordinates (cos 45°, sin 45°).A card reads 405°, a lap and 45° more. What is its cosine?
A card reads -90°. Which angle between 0° and 360° lands on the same point? Type the number.
WHY THIS EXERCISEEvery card is the same move. Add or subtract full turns, then read the point.
Draw the family circle with the eight card angles marked and the exact coordinates at each point.
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