証明しようとする式:
    \[ \frac{\overline{ OP'} - r }{ \Delta t} \ \ \ \longrightarrow \ v\ cos( \tau ) \ \ \ ( \Delta t \longrightarrow 0 ) \\ \ \\ \frac{ 1 - cos( a ) }{\Delta t } \ \ \ \longrightarrow \ 0 \ \ \ ( \Delta t \longrightarrow 0 ) \\ \ \\ \frac{ sin( a ) }{\Delta t } \ \ \longrightarrow \ \frac{ v\ sin(\tau) }{r} \ \ \ \ ( \Delta t \longrightarrow 0 ) \]


    \( \overline{ OP'} \) は, \[ \overline{ OP'}^2 \\ = ( r + v\ \Delta t\ cos( \tau ) )^2 \\ \quad + ( v\ \Delta t\ sin( \tau ) )^2 \\ = r^2 + 2\ r\ v\ \Delta t\ cos( \tau ) + v^2 \Delta t^2\ cos( \tau )^2 \\ \quad + v^2 \Delta t^2\ sin( \tau )^2 \\ = r^2 + 2\ r\ v\ \Delta t\ cos( \tau ) + v^2 \Delta t^2 \\ \] よって, \[ \frac{\overline{ OP'}^2 - r^2 }{ \Delta t} \\ = 2 r v\ cos( \tau ) + v^2 \Delta t \ \\ \ \\ \longrightarrow \ 2 r v\ cos( \tau ) \ \ ( \Delta t \longrightarrow 0 ) \\ \ \\ \ \\ \frac{\overline{ OP'} - r }{ \Delta t} \\ = \frac{\overline{ OP'}^2 - r^2 }{ \Delta t} \ \ \frac{ 1 }{\overline{ OP'} + r } \ \\ \ \\ \longrightarrow \ v\ cos( \tau ) \ \ ( \Delta t \longrightarrow 0 ) \] また, \[ \frac{ 1 - cos( a ) }{\Delta t } \\ \ \\ = \frac{1}{ \Delta t } \frac{ \overline{ OP'} - \overline{ OP'} cos( a ) }{ \overline{ OP'} } \\ \ \\ = \frac{1}{\overline{ OP'} \Delta t } \bigl( \overline{ OP'} - ( r + v \Delta t cos( \tau ) ) \bigr) \\ \ \\ = \frac{1}{\overline{ OP'}} \bigl( \frac{ \overline{ OP'} - r }{\Delta t} - v cos( \tau ) \bigr) \\ \ \\ \longrightarrow \ \frac{ v\ cos( \tau ) - v cos( \tau ) }{ r } \ \ \ \ = 0 \ \ ( \Delta t \longrightarrow 0 ) \]
    \[ \frac{ sin( a ) }{\Delta t } \\ \ \\ = \frac{1}{ \Delta t }\ \frac{ \overline{ OP'}\ sin( a ) }{ \overline{ OP'} } \\ \ \\ = \frac{1}{\overline{ OP'} \Delta t }\ v\ \Delta t\ sin(\tau) \\ \ \\ \longrightarrow \ \frac{ v\ sin(\tau) }{r} \ \ ( \Delta t \longrightarrow 0 ) \]


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