Examples

1. In the circle $\rho = r\sin\, \theta$, find $\psi$ and $\tau$ in terms of $\theta$.

   Solution: $\psi = \theta$, $\tau=2\theta$.

2. In the parabola $\rho = a \sec^ \frac{\theta}{2}$, show that $\tau + \psi = \pi$.

3. In the curve $\rho^2 = a^2\cos\, 2\theta$, show that $2\psi = \pi + 4\theta$.

4. Show that $\psi$ is constant in the logarithmic spiral $\rho = e^{a\theta}$. Since the tangent makes a constant angle with the radius vector, this curve is also called the equiangular spiral.

5. Given the curve $\rho = a \sin^3 \frac{\theta}{3}$, prove that $\tau = 4\psi$.

   SAGE can help with this problem. Using (6.12) but with $t$ in place of $\theta$ for typographical simplicity, we have

   [fontsize=\small,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
   
   sage: a,t = var("a,t")
   sage: r = a*sin(t/3)^3
   sage: tanpsi = r/diff(r,t); tanpsi
   sin(t/3)/cos(t/3)
   

   Therefore, $\tan(\psi) = \tan(\theta/3)$, so $\theta=3\psi$. Therefore, according to (6.13), we have $\tau=\theta+\psi=3\psi+\psi=4\psi$, as expected.

6. Show that $\tan\, \psi = \theta$ in the spiral of Archimedes $\rho = a\theta$. Find values of $\psi$ when $\theta = 2\pi$ and $4\pi$.

   Solution: $\psi = 80^o 57' = 1.4128...$ and $85^o 27'=1.4913...$.

7. Find the angle between the straight line $\rho\cos\,\theta = 2a$ and the circle $\rho = 5a\sin\,\theta$.

   Solution: $\arctan \frac{3}{4}$.

8. Show that the parabolas $\rho = a \sec^2 \frac{\theta}{2}$ and $\rho = b \csc^2 \frac{\theta}{2}$ intersect at right angles.

9. Find the angle of intersection of $\rho = a\sin\,\theta$ and $\rho = a\sin\, 2\theta$.

   Solution: At origin $0^o$; at two other points $\arctan 3 \sqrt{3}$.

10. Find the slopes of the following curves at the points designated:

    curve	point	solution (if given)
    (a) $\rho = a(l - \cos,\theta)$	$\theta = \frac{\pi}{2}$	$-1$
    (b) $\rho = a\sec^2 \theta$	$\rho = 2a$	$3$
    (c) $\rho = a\sin\, 4\theta$	origin	$0, 1, \infty, -1$
    (d) $\rho^2 = a^2\sin\, 4\theta$	origin	$0, 1, \infty, -1$
    (e) $\rho = a\sin\, 3\theta$	origin	$0, \sqrt{3}, -\sqrt{3}$
    (f) $\rho = a\cos\, 3\theta$	origin
    (g) $\rho = a\cos\, 2\theta$	origin
    (h) $\rho = a\sin\, 2\theta$	$\theta = \frac{\pi}{4}$
    (i) $\rho = a\sin\, 3\theta$	$\theta = \frac{\pi}{6}$
    (j) $\rho = a\theta$	$\theta = \frac{\pi}{2}$
    (k) $\rho\theta = a$	$\theta = \frac{\pi}{2}$
    (l) $\rho = e^\theta$	$\theta = 0$

11. Prove that the spiral of Archimedes $\rho = a\theta$, and the reciprocal spiral $\rho = \frac{a}{\theta}$, intersect at right angles.

12. Find the angle between the parabola $\rho = a \sec^2 \frac{\theta}{2}$ and the straight line $\rho\sin\, \theta = 2a$.

    Solution: $45^o=\pi/4$.

13. Show that the two cardioids $\rho = a(1 + \cos\, \theta)$ and $\rho = a(1 - \cos\, \theta)$ cut each other perpendicularly.

14. Find lengths of subtangent, subnormal, tangent, and normal of the spiral of Archimedes $\rho = a\theta$.

    Solution: subt. = $\frac{\rho^2}{a}$, tan. = $\frac{\rho}{a} \sqrt{a^2 + \rho^2}$, subn. = $a$, nor. = $\sqrt{a^2 + \rho^2}$. The student should note the fact that the subnormal is constant.

15. Get lengths of subtangent, subnormal, tangent, and normal in the logarithmic spiral $\rho = a^\theta$.

    Solution: subt. = $\frac{\rho}{\log a}$, tan. = $\rho \sqrt{1 + \frac{1}{\log^2 a}}$, subn. = $\rho\log\, a$, nor. = $\rho \sqrt{1 + \log^2 a}$.

    When $a = e$, we notice that subt. = subn., and tan. = nor.

16. Find the angles between the curves $\rho = a(1 + \cos\, \theta)$ and $\rho = b(1 - \cos\, \theta)$.

    Solution: 0 and $\frac{\pi}{2}$.

17. Show that the reciprocal spiral $\rho = \frac{a}{\theta}$ has a constant subtangent.

18. Show that the equilateral hyperbolas $\rho^2\sin\, 2\theta=a^2$, $\rho^2\cos\, 2\theta=b^2$ intersect at right angles.