• lab{10.6.1a} For each of the following values of [math]\theta[/math], find the value of [math]r[/math] such that [math]r=4 \sin \theta[/math]:
  [[math]] \theta = 0, \frac\pi6, \frac\pi4, \frac\pi2, \frac{3\pi}4, \frac{5\pi}6, \pi . [[/math]]
• Plot the seven points with the polar coordinates [math](r,\theta)[/math] found in part \ref{ex10.6.1a}.
• Draw and identify the curve defined by the equation [math]r=4\sin\theta[/math] in polar coordinates.

• lab{10.6.2a} For each of the following values of [math]\theta[/math], find the value of [math]r[/math] such that [math]r = 2(1+\cos \theta)[/math]:
  [[math]] \theta = 0, \frac\pi4, \frac\pi3, \frac\pi2, \frac{2\pi}3, \frac{5\pi}6, \pi . [[/math]]
• Plot the seven points with the polar coordinates [math](r,\theta)[/math], found in part \ref{ex10.6.2a}.
• lab{10.6.2c} What symmetry property is possessed by the curve defined by the equation [math]r=2(1+\cos\theta)[/math] in polar coordinates?
• Draw the curve in part \ref{ex10.6.2c}.

Using a figure and the geometric interpretation of polar coordinates, show that [math]r = \frac5{\cos\theta}[/math] is an equation in polar coordinates of the vertical line cutting the [math]x[/math]-axis in the point [math](5,0)[/math].

Using a figure and the geometric interpretation of polar coordinates, find an equation in polar coordinates of the horizontal line cutting the [math]y[/math]-axis in the point [math](0,5)[/math].

Assume the well-known fact that, if one side of a triangle inscribed in a circle is a diameter, then the triangle is a right triangle. Using this fact and the geometric interpretation of polar coordinates, show that [math]\cos\theta = \frac{r}{2a}[/math] is an equation of the circle which passes through the origin and has radius [math]a \gt 0[/math] and center on the [math]x[/math]-axis.

Identify and draw the polar graphs of the two equations

• [math]r=7[/math]
• [math]\theta = \frac\pi6[/math].

Consider the curves defined by each of the following equations in polar coordinates. Write each curve as the graph of an equation in [math]x[/math]- and [math]y[/math]-coordinates. Identify and draw the curve in the [math]xy[/math]-plane.

• [math]r\cos\theta = -2[/math]
• [math]r\sin\theta = 4[/math]
• [math]r=-4\cos\theta[/math]
• [math]r=\frac2{\sin\theta-2\cos\theta}[/math]
• [math]r=\frac1{1-\cos\theta}[/math] (see Example \ref{exam 10.6.3})
• [math]r=5[/math]
• [math]\theta = \arcsin \frac3{\sqrt{10}}[/math]
• [math]r = \frac1{2-\sqrt3 \cos\theta}[/math].

Let [math]f[/math] be a real-valued function of a real variable. Prove that:

• If [math]f[/math] is an even function, then the polar graph of the equation [math]r=f(\theta)[/math] is symmetric about the [math]x[/math]-axis.
• If [math]f[/math] is an odd function, then the polar graph of the equation [math]r = f(\theta)[/math] is symmetric about the [math]y[/math]-axis.

Let [math]F[/math] be a real-valued function of two real variables. Prove that the polar graph of the equation [math]F(r^2,\theta) = 0[/math] is symmetric about the origin.

Draw the curve defined by each of the following equations in polar coordinates (the number [math]a[/math] is an arbitrary positive constant).

• [math]r=a(1+\cos\theta)[/math] (a cardioid).
• [math]r=a(2+\cos\theta)[/math] (a lima\c{con}).
• [math]r=a(\frac12 + \cos\theta)[/math] (a lima\c{con}).
• [math]r^2 = 2a^2\sin2\theta[/math] (a lemniscate).
• [math]r\theta = 2[/math] (a hyperbolic spiral).