b_{y} - b_{x}a_{y}|}{|a||b|} = \frac{|\vec{a} \times \vec{b}|}{|a||b|}
$$</p>
<p>Calculating \(\cos(\beta)\) and \(\sin(\beta)\):</p>
<p>$$
\cos(\beta) = \frac{\vec{c} \cdot \vec{d}}{|c||d|} = \frac{c_{x}d_{x} + c_{y}d_{y}}{|c||d|}
$$</p>
<p>$$
\sin(\beta) = \frac{|\vec{c} \times \vec{d}|}{|c||d|} = \frac{|c_{x}d_{y} - d_{x}c_{y}|}{|c||d|}
$$</p>
<p>Final Equation:</p>
<p>$$
\sin(\alpha + \beta) = \frac{|a_{x}b_{y} - b_{x}a_{y}|\cdot(c_{x}d_{x} + c_{y}d_{y}) + (a_{x}b_{x} + a_{y}b_{y})\cdot|c_{x}d_{y} - c_{