UNIT 3 SAMPLE WORK TRIGONOMETRY PILE UP
Solve for X

First of all, yuck.

• Start from the bottom, in ∆1:

sin(30°) = 4 / (a₁ + 2.5)   →   a₁ = 4 / sin(30°) - 2.5 = 5.5

where a₁ is the part of the hypotenuse of ∆1 that overlaps with the hypotenuse of ∆2.

• In ∆2:

cos(15°) = (2.2 + a₂) / (3.5 + a₁)   →   a₂ = (3.5 + a₁) cos(15°) - 2.2 ≈ 6.493

where a₂ is the part of the leg of ∆2 that overlaps with the hypotenuse of ∆3.

• In ∆3:

a₃² + 3.5² = (a₂ + 2.4)²   →   a₃ = √((a₂ + 2.4)² - 3.5²) ≈ 8.176

where a₃ is the leg of ∆3 that overlaps with ∆4 and ∆5.

• In ∆4:

tan(51°) = a₄ / 2.855   →   a₄ = 2.855 tan(51°) ≈ 3.526

where a₄ is the leg of ∆4 that overlaps with the leg of ∆5.

• In ∆5:

The overlapping legs of ∆4 and ∆5 have the same length, so

sin(48°) = a₄ / a₅   →   a₅ = a₄ / sin(48°) ≈ 4.744

where a₅ is the hypotenuse of ∆5.

• In ∆9:

sin(58°) = (a₆ + 2.5) / (4.18 + a₅)   →   a₆ = (4.18 + a₅) sin(58°) - 2.5 ≈ 5.068

where a₆ is the leg of ∆10 that overlaps with the leg of ∆9.

• In ∆10:

a₇² = 2.2² + a₆²   →   a₇ = √(2.2² + a₆²) ≈ 5.525

where a₇ is the hypotenuse of ∆10.

• In ∆11:

cos(21°) = a₇ / a₈   →   a₈ = a₇ / cos(21°) ≈ 5.918

where a₈ is the hypotenuse of ∆11.

• In ∆12: (finally!)

sin(74°) = x / (a₈ + 2.6)   →   x = (a₈ + 2.6) sin(74°) ≈ 8.19

the two may have that, even a breakdown is probably offered