Footbridge Cables: Principal Arctan and the Missing 180 Degrees
A cable-stayed mast has two straight cables anchored at the same point on a horizontal deck and rising on opposite sides of the anchor. The acute angles the cables make with the deck are α and β, and a surveyor records tan α = 2 and tan β = 3. The design angle is the upward angle between the two cables, α + β. The design office uses the identity tan(α + β) = (tan α + tan β)/(1 − tan α tan β), obtains −1, and then applies the principal value arctan, which returns values in (−90°, 90°), to report the design angle as −45°. The site engineer objects that an angle between two cables cannot be negative and that the connector is specified for angles below 120°. A second design variant has tan α = 2 and tan β = 1/3. The office wants a rule for when the principal arctan answer is safe and when a correction of +180° is required.
1. For tan α = 2 and tan β = 3, find α + β exactly and explain why the principal arctan gives −45°.
Read the worked answer
tan(α + β) = (2 + 3)/(1 − 2·3) = 5/(−5) = −1.
Since α and β are acute, 0 < α, β < π/2, so 0 < α + β < π. Because tan(α + β) is negative, α + β must lie in (π/2, π), not in (−π/2, 0). The principal arctan of −1 is −π/4, which lies in (−π/2, π/2). The angle in (π/2, π) with the same tangent is −π/4 + π = 3π/4 = 135°. Thus α + β = 135°, and the spreadsheet's −45° is the principal value, not the physical design angle.
2. Does the first design meet the connector specification of an angle below 120°? Justify.
Read the worked answer
The actual design angle is 135°, which is greater than 120°. Therefore the first design does not meet the connector specification. The erroneous −45° report would wrongly suggest that it passes, because −45° is below 120°. The branch correction therefore changes the engineering decision, not just the display.
3. For the second variant tan α = 2 and tan β = 1/3, compute α + β and state whether a +180° correction is needed. Give a general rule for acute α and β.
Read the worked answer
tan(α + β) = (2 + 1/3)/(1 − 2·(1/3)) = (7/3)/(1/3) = 7.
Here tan α tan β = 2/3 < 1. For acute α and β this makes α + β < π/2, so the principal arctan 7 gives the correct angle directly: α + β ≈ 81.87°, with no +180° correction.
General rule for acute α and β: compute P = tan α tan β. If P < 1, then α + β lies in (0, π/2), and the principal arctan is correct. If P > 1, then α + β lies in (π/2, π), so add 180° to the principal arctan. If P = 1, then tan(α + β) is undefined and α + β = 90°.