# How do you find the points of horizontal tangency of r=3cos2thetasectheta?

Dec 31, 2017

$\theta = \pm {25.92}^{\circ}$

#### Explanation:

Let us find $\frac{\mathrm{dy}}{\mathrm{dx}}$ in terms of $r$ and $\theta$ using $x = r \cos \theta$ and $y = r \sin \theta$.

As such $\frac{\mathrm{dx}}{d \theta} = \frac{\mathrm{dr}}{d \theta} \cos \theta - r \sin \theta$

and $\frac{\mathrm{dy}}{d \theta} = \frac{\mathrm{dr}}{d \theta} \sin \theta + r \cos \theta$

i.e. $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dr}}{d \theta} \sin \theta + r \cos \theta}{\frac{\mathrm{dr}}{d \theta} \cos \theta - r \sin \theta}$

As $r = 3 \cos 2 \theta \sec \theta$

$\frac{\mathrm{dr}}{d \theta} = - 6 \sin 2 \theta \sec \theta + 3 \cos 2 \theta \sec \theta \tan \theta$

and $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$, when

$\sin \theta \left(3 \cos 2 \theta \sec \theta \tan \theta - 6 \sin 2 \theta \sec \theta\right) + r \cos \theta = 0$

or $3 \cos 2 \theta {\tan}^{2} \theta - 12 {\sin}^{2} \theta + r \cos \theta = 0$

or $r = \frac{12 {\sin}^{2} \theta - 3 \cos 2 \theta {\tan}^{2} \theta}{\cos \theta}$

or $r = 12 {\sin}^{2} \theta \sec \theta - 3 \cos 2 \theta \sec \theta {\tan}^{2} \theta$

or $r = 12 {\sin}^{2} \theta \sec \theta - r {\tan}^{2} \theta$

i.e. $r = \frac{12 {\sin}^{2} \theta \sec \theta}{1 + {\tan}^{2} \theta} = \frac{2 \tan \theta}{1 + {\tan}^{2} \theta} \cdot 6 \sin \theta$

= $6 \sin 2 \theta \sin \theta$

or $3 \cos 2 \theta \sec \theta = 6 \sin 2 \theta \sin \theta$

or $\cos 2 \theta = {\sin}^{2} 2 \theta$

or $\cos 2 \theta = 1 - {\cos}^{2} 2 \theta$

or ${\cos}^{2} 2 \theta + \cos 2 \theta - 1 = 0$

or $\cos 2 \theta = \frac{- 1 + \sqrt{5}}{2} = 0.618$

Observe that we cannot have cos2theta=(-1-sqrt5)/

or $2 \theta = \pm {51.83}^{\circ}$

or $\theta = \pm {25.92}^{\circ}$

i.e. at $y = r \sin \theta = \pm 3 \cos {51.83}^{\circ} \sec {25.92}^{\circ} \sin {25.92}^{\circ}$

= $3 \times 0.618 \times 1.112 \times 0.437 = 0.9$

The graph of curve is shown below:

graph{(x^3+xy^2-3x^2+3y^2)(y^2-0.81)=0 [-5, 5, -2.5, 2.5]}