Efficient Calculation of Final Magnitude and Angle in Vector Addition Method

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okay this is what I can get one my own

$$ \left( \sum (L_n cos(\theta_n - \theta_t)) \right)^2 = \left( \sum (L_n sin (\theta_n)) \right)^2 + \left( \sum (L_n cos(\theta_n)) \right)^2 $$

$$ \left( \sum (L_n cos(\theta_n - \theta_t)) \right)^2 / \left( \sum (L_n) \right)^2 = \left( \left( \sum (L_n sin (\theta_n)) \right)^2 + \left( \sum (L_n cos(\theta_n)) \right)^2 \right) / \left( \sum (L_n) \right)^2 $$

$$ \left( \sum (cos(\theta_n) cos(\theta_t) + sin(\theta_n) sin(\theta_t)) \right)^2 = \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 $$

$$ \left( \sum (cos(\theta_n) cos(\theta_t)) + \sum (sin(\theta_n) sin(\theta_t)) \right)^2 = \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 $$

$$ \left( cos(\theta_t) \sum (cos(\theta_n)) + sin(\theta_t) \sum (sin(\theta_n)) \right)^2 = \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 $$

$$ \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 +2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) + \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 = \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 $$
 
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$$ \left( \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 +2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) + \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) = \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) $$

$$ \left( \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) + \left( 2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) + \left( \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 \right) / \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) = 1 $$

$$ \left( cos(\theta_t) \right)^2 / \left( \left( \sum (sin (\theta_n)) \right)^2 + 1 \right) + \left( 2 cos(\theta_t) sin(\theta_t) \right) / \left( \left( \sum (sin (\theta_n)) \right) + \left( \sum (cos(\theta_n)) \right) \right) + \left( sin(\theta_t) \right)^2 / \left( 1 + \left( \sum (cos(\theta_n)) \right)^2 \right) = 1 $$
 
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$$ \left( \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 + 2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) + \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) = \left( \left( \sum (sin (\theta_n)) \right)^2 + \left( \sum (cos(\theta_n)) \right)^2 \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) $$

$$ \left( \left( cos(\theta_t) \sum (cos(\theta_n)) \right)^2 \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) + \left( 2 cos(\theta_t) \sum (cos(\theta_n)) sin(\theta_t) \sum (sin(\theta_n)) \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) + \left( \left( sin(\theta_t) \sum (sin(\theta_n)) \right)^2 \right) / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) = \left( \sum (sin (\theta_n)) \right)^2 / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) + \left( \sum (cos(\theta_n)) \right)^2 / \left( \sum (sin (\theta_n)) \sum (cos(\theta_n)) \right) $$

$$ \left( cos(\theta_t) \right)^2 \sum (cot(\theta_n)) + 2 cos(\theta_t) sin(\theta_t) + \left( sin(\theta_t) \right)^2 \sum (tan(\theta_n)) = \sum (tan (\theta_n)) + \sum (cot(\theta_n)) $$
 
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found an error in my equation, this is fixed

$$ \frac {\left( cos(\theta_t) \sum (L_n cos(\theta_n)) \right)^2 + 2 cos(\theta_t) \sum (L_n cos(\theta_n)) sin(\theta_t) \sum (L_n sin(\theta_n)) + \left( sin(\theta_t) \sum (L_n sin(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} = \frac {\left( \sum (L_n sin (\theta_n)) \right)^2 + \left( \sum (L_n cos(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} $$

$$ \frac {\left( cos(\theta_t) \sum (L_n cos(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} + \frac {2 cos(\theta_t) \sum (L_n cos(\theta_n)) sin(\theta_t) \sum (L_n sin(\theta_n))} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} + \frac {\left( sin(\theta_t) \sum (L_n sin(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} = \frac {\left( \sum (L_n sin (\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} + \frac {\left( \sum (L_n cos(\theta_n)) \right)^2} {\sum (L_n sin (\theta_n)) \sum (L_n cos(\theta_n))} $$

$$ cos ^2 (\theta_t) \frac {\sum L_n cos(\theta_n)} {\sum L_n sin (\theta_n)} + 2 cos(\theta_t) sin(\theta_t) + sin ^2 (\theta_t) \frac {\sum L_n sin(\theta_n)} {\sum L_n cos(\theta_n)} = \frac {\sum L_n sin (\theta_n)} {\sum L_n cos(\theta_n)} + \frac {\sum L_n cos(\theta_n)} {\sum L_n sin (\theta_n)} $$
 
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