A spelunker is surveying a cave. She follows a passage 180 m straight west, then 290 m in a direction 45∘ east of south, and then 280 m at 30∘ east of north. After a fourth unmeasured displacement, she finds herself back where she started. Determine the direction of the fourth displacement? (degrees) (SOUTH OF WEST)

The direction of the fourth displacement is θ = 12.775 (south of west)

Explanation:

write down the values in the question,

The passage is 180m west, 290m in the direction 45° east of south, 280m at 30° east of north.

x = 290 * cos 45° + 280 * sin 30° - 180

x=165.06 m

y = 280 cos (30° - 290 * sin (45°)

y = 37.426°

θ = tan ^-1 (y/x) = tan^-1 (37.426 / 165.06)

θ = 12.775° (south of west)