Friday morning the Boathouse wall holds the week: two triangles, one circle and a table of twelve angles.
"tan 45° is 1. tan 30° is √3/3. tan 60° is √3," Comet reads. "Everything else is a reflection or a turn."
"What do you notice about the signs?" Wren asks.
"They come from the quadrant," Comet says. "The size comes from the reference angle."
Nova sorts the angles into four groups. "Would you like a hint? One group for each quadrant."
"The same three sizes, four times," Wren says. "Today we read all twelve without slowing down."
"And then the ramp," Comet says. "It rises 1 meter over a run of √3 meters. What angle is that?"
| Question | Ratio or rule | Answer |
|---|---|---|
| The light's angle when the 4 m mast casts a 4 m shadow | tan = 4 ÷ 4 = 1 | 45° |
| The light's angle for a 6.9 m shadow | tan ≈ 0.58 ≈ √3/3 | 30° |
| cos 150° | cos(π - x) = -cos x | -√3/2 |
| tan 210° | tan(π + x) = tan x | √3/3 |
| sin 390° | a full turn changes nothing | 1/2 |
Every row has the same shape: a special angle, a reflection or a turn, and a sign from the quadrant.
The ramp question is new. Rise over run is tangent, so a rise of 1 over a run of √3 is tan 30°.
A line through the origin at angle x has slope tan x, because slope is rise over run.
A 45° ramp has slope 1. A 30° ramp has slope √3/3 ≈ 0.58. A 60° ramp has slope √3 ≈ 1.73.
At 90° the run is zero, so tan 90° has no value. The tangent graph has a wall there, like week 3's asymptotes.
A classmate says tan 120° = √3. Check the quadrant: 120° is in quadrant 2, where sine is positive and cosine negative.
So tangent is negative there, and tan 120° = -√3. The size was right; the sign came from the quadrant.
| Week review | True or false? |
|---|---|
| Tangent is sine over cosine. | ? |
| tan 90° = 1. | ? |
| sin(π - x) = sin x and cos(2π - x) = cos x. | ? |
| Cosine is odd. | ? |
| The slope of a line at angle x through the origin is tan x. | ? |
A full week of exact values. Tomorrow is Dock Day, and a shadow at home gives up its angle.