(cos theta + i sin theta)^10/(sin theta + i cos theta)^10 simplify this

asked by guest
on Nov 25, 2024 at 4:12 am



You asked:

Simplify the expression: \(\frac{{\left( \cos\left( θ \right) + i \cdot \sin\left( θ \right) \right)}^{10}}{{\left( \sin\left( θ \right) + i \cdot \cos\left( θ \right) \right)}^{10}}\)

MathBot Answer:

Simplified:

\[ \frac{{\left( \cos\left( θ \right) + i \cdot \sin\left( θ \right) \right)}^{10}}{{\left( \sin\left( θ \right) + i \cdot \cos\left( θ \right) \right)}^{10}} = \frac{e^{10 θ i}}{\left(\sin{\left(θ \right)} + i \cos{\left(θ \right)}\right)^{10}} \]

Expanded:

\[ \frac{{\left( \cos\left( θ \right) + i \cdot \sin\left( θ \right) \right)}^{10}}{{\left( \sin\left( θ \right) + i \cdot \cos\left( θ \right) \right)}^{10}} = - \frac{\sin^{10}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} + \frac{10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} + \frac{45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} - \frac{120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} - \frac{210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} + \frac{252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} + \frac{210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} - \frac{120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} - \frac{45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} + \frac{10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} + \frac{\cos^{10}{\left(θ \right)}}{\sin^{10}{\left(θ \right)} + 10 i \sin^{9}{\left(θ \right)} \cos{\left(θ \right)} - 45 \sin^{8}{\left(θ \right)} \cos^{2}{\left(θ \right)} - 120 i \sin^{7}{\left(θ \right)} \cos^{3}{\left(θ \right)} + 210 \sin^{6}{\left(θ \right)} \cos^{4}{\left(θ \right)} + 252 i \sin^{5}{\left(θ \right)} \cos^{5}{\left(θ \right)} - 210 \sin^{4}{\left(θ \right)} \cos^{6}{\left(θ \right)} - 120 i \sin^{3}{\left(θ \right)} \cos^{7}{\left(θ \right)} + 45 \sin^{2}{\left(θ \right)} \cos^{8}{\left(θ \right)} + 10 i \sin{\left(θ \right)} \cos^{9}{\left(θ \right)} - \cos^{10}{\left(θ \right)}} \]

Factored:

\[ \frac{{\left( \cos\left( θ \right) + i \cdot \sin\left( θ \right) \right)}^{10}}{{\left( \sin\left( θ \right) + i \cdot \cos\left( θ \right) \right)}^{10}} = - \frac{\left(i \sin{\left(θ \right)} + \cos{\left(θ \right)}\right)^{10}}{\left(- i \sin{\left(θ \right)} + \cos{\left(θ \right)}\right)^{10}} \]


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