rewrite each equation in rectangular form and describe the graph. solve for y if possible.

a. theta = pi/6
b. r=8costheta

θ = π/6

tanθ = 1/√3
y/x = 1/√3
y = 1/√3 x

r = 8cosθ
r^2 = 8r cosθ
x^2+y^2 = 8x

To rewrite each equation in rectangular form, we'll need to convert them from polar form to rectangular form. The rectangular form is given by (x, y), where x and y are Cartesian coordinates.

a. For the equation theta = pi/6, we can rewrite it in rectangular form using the following formulas:
x = r * cos(theta)
y = r * sin(theta)

In this case, theta = pi/6, so the equation becomes:
x = r * cos(pi/6)
y = r * sin(pi/6)

Since no value for r is given, we can't determine the exact values of x and y. However, we can simplify further by substituting the specific value of pi/6.

Using the values of cos(pi/6) = sqrt(3)/2 and sin(pi/6) = 1/2, the equation becomes:
x = r * (sqrt(3)/2)
y = r * (1/2)

b. For the equation r = 8cos(theta), we again use the formulas for rectangular form conversion:
x = r * cos(theta)
y = r * sin(theta)

Now, we substitute the given equation, r = 8cos(theta), into the formulas:
x = (8cos(theta)) * cos(theta)
y = (8cos(theta)) * sin(theta)

We can simplify further using the trigonometric identity cos^2(theta) + sin^2(theta) = 1:
x = 8cos^2(theta)
y = 8cos(theta) * sin(theta)

To solve for y, we need more information or values for theta. Without specific values, it is not possible to determine the exact value of y.

Regarding the graph of each equation:
a. For the equation theta = pi/6, it represents a single point in the polar coordinate system. In the rectangular coordinate system, it would correspond to a point on the x-axis. The graph would show a single point at (x, y), where x and y values are determined by multiplying the radius r (whose value is missing) by the trigonometric functions of pi/6.

b. For the equation r = 8cos(theta), the graph would represent a cardioid shape in the polar coordinate system. In the rectangular coordinate system, it would resemble a heart. Again, the exact shape and position on the graph would depend on the values of theta and the missing radius r.