proof involving polar curve :)

[tlk2]

Hi, I can't figure out how to continue on this problem. Any help with reasons would be appreciated. Thanks!

P any point except origin on curve r=f(theta)

? is the angel between the tangent line at P and the radial line OP

show that tan?=r/(dr/d theta ) (hint- ?=c-theta, where c is the angle where the tangent line meets the horizontal line)

I've got:
?=c-theta
tan?=tan(c-theta)=(tanc-tan theta)/(1+tanc tan theta)
Can't figure out where to go from here...

 

[galactus]

Are you asking to prove that the tangent of the angle between the radial line and the tangent line at the point P is given by

\(\displaystyle tan({\alpha})=\frac{r}{\frac{dr}{d\theta}}\)?.

The tangent line intersects the polar axis at Q. Then, the slope of the tangent line through Pand Q is

\(\displaystyle \frac{fcos{\theta}+f'sin{\theta}}{-fsin{\theta}+f'cos{\theta}}\)...........[1]

This is equal to:

\(\displaystyle tan({\theta}+{\alpha})=\frac{tan{\theta}+tan{\alpha}}{1-tan{\theta}tan{\alpha}}\)

\(\displaystyle =\frac{sin{\theta}+cos{\theta}tan{\alpha}}{cos{\theta}-sin{\theta}tan{\alpha}}\)...........[2]

equate [1] and [2], cross multiply:

\(\displaystyle (fcos{\theta}+f'sin{\theta})(cos{\theta}-sin{\theta}tan{\alpha})=(sin{\theta}+cos{\theta}tan{\alpha})(-fsin{\theta}+f'cos{\theta})\)

see if you can hammer it into shape.