Answer:
190.79 pounds at 88.11° or 191.01 pounds at 89.40°
Step-by-step explanation:
The effect of rounding will be different, depending on the method you choose for solving the problem. In any event, rounding intermediate results is an inappropriate way to solve a problem like this.
We believe we can minimize the effect of rounding by using the law of cosines to find the magnitude.
f = f1 +f2
|f|² = |f1|² +|f2|² -2|f1|·|f2|·cos(180°-(170° -60°))
|f|² = 200² +100² -2·200·100·cos(70°) ≈ 40,000 +10,000 -40,000·(0.34)
|f|² ≈ 36,400
|f| ≈ 190.79 . . . . . due to rounding cos(70°). Should be 190.576.
The angle of f can be computed from the law of sines. We choose to compute the angle of the summation triangle that is opposite f1. Call it ∠1. Then we have ...
sin(∠1)/|f1| = sin(70°)/|f|
∠1 = arcsin(|f1|·sin(70°)/|f|) = arcsin(200·sin(70°)/190.79)
∠1 = arcsin(200·0.94/190.79) = arcsin(188/190.79) = arcsin(0.99)
∠1 = 81.89°
To find the resultant angle, this angle is subtracted from 170°:
170° -81.89° = 88.11°
The direction of the resultant is 88.11° and its magnitude is 190.79 pounds.
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Using appropriate end-of-calculation rounding, the resultant would be calculated as 190.56∠89.54° pounds.
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If we compute the result by decomposing into horizontal and vertical components, we expect the effects of rounding to be far worse.
200 cis 60° + 100 cis 170° = 200(cos(60°), sin(60°)) +100(cos(170°), sin(170°))
= 200(0.50, 0.87) +100(-0.98, 0.17) = (100, 174) +(-98, 17) = (2, 191)
Then the magnitude is ...
|f| = √(2² +191²) = √36485 ≈ 191.01
and the angle is ...
∠f = arctan(191/2) = arctan(95.50) = 89.40°
Calculated in this way, the resultant is ...
direction: 89.40°, magnitude: 191.01 pounds
