1. One way to approach this is to convert the magnitude and direction of each force to rectangular coordinates.
... first force = 90(cos(30°), sin(30°)) = (77.94, 45.00)
... second force = 50(cos(160°), sin(160°)) = (-46.98, 17.10)
Then their sum is
... resultant force = first force + second force = (30.96, 62.10)
The magnitude is found using the Pythagorean theorem:
... resultant magnitude = √(30.96^2 + 62.10^2)
... ... = √(958.52+3856.41) = √4814.93
... ... = 69.39
The resultant direction can be found using the definition of the tangent function.
... tan(α) = 62.10/30.96 = 2.01
... α = arctan(2.01) = 63.55°
The (magnitude, direction) of the resultant force are (69.39, 63.55°).
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(Note that rounding introduced an error of 0.05° in the direction.)
2. Using DeMoivre's theorem, the magnitude of the 4th root is the 4th root of the magnitude, and the angle of the 4th root is 1/4 of the angle (and its aliases).
... fourth root = 256^(1/4)∠((240°+k*360°)/4) = 4∠(60°+k*90°) . . . . k=0, 1, 2, 3
