1.Write the rectangular equation in polar form.

(x – 4)2 + y2 = 16

2.Write the equation –2x + 6y = 7 in polar form.

3.Graph the polar equation 2 = r cos(theta – 20°).

4. Write the polar equation in rectangular form.
r = –12 cos

1. To convert the given rectangular equation (x - 4)^2 + y^2 = 16 into polar form, we need to substitute x and y with their respective polar representations.

Let's consider that x = r cos(theta) and y = r sin(theta), where r is the radial distance from the origin to the point (x, y), and theta is the angle formed with the positive x-axis.

So, (x - 4)^2 + y^2 = 16 becomes (r cos(theta) - 4)^2 + (r sin(theta))^2 = 16.

Expanding and simplifying, we get r^2 cos^2(theta) - 8r cos(theta) + 16 + r^2 sin^2(theta) = 16.

Since cos^2(theta) + sin^2(theta) = 1, the equation further simplifies to r^2 - 8r cos(theta) + 16 = 0, which is the polar form of the rectangular equation.

2. To convert the equation -2x + 6y = 7 into polar form, let's again use the substitution x = r cos(theta) and y = r sin(theta).

Substituting these values into the given equation, we have -2(r cos(theta)) + 6(r sin(theta)) = 7.

Dividing both sides by r, we get -2 cos(theta) + 6 sin(theta) = 7/r.

This is the polar form of the equation -2x + 6y = 7.

3. To graph the polar equation 2 = r cos(theta - 20°), we need to plot points for different values of theta and find the corresponding r values.

Start by considering various values of theta, such as 0°, 30°, 60°, 90°, and so on. Calculate the corresponding r values using r = 2 / cos(theta - 20°).

For example, when theta = 0°, r = 2 / cos(0° - 20°) = 2 / cos(-20°) ≈ 2.495.

Repeat this process for different values of theta and plot the points (r, theta) on a polar coordinate system. Connect these points to form a smooth curve to represent the graph of the polar equation.

4. To convert the polar equation r = -12 cos(theta) into rectangular form, we use the equations x = r cos(theta) and y = r sin(theta).

Substituting r = -12 cos(theta), we get x = (-12 cos(theta)) cos(theta) and y = (-12 cos(theta)) sin(theta).

Simplifying further, we have x = -12 cos^2(theta) and y = -12 cos(theta) sin(theta).

By applying the trigonometric identity sin(2theta) = 2sin(theta)cos(theta), we can rewrite y as y = -6sin(2theta).

Therefore, the polar equation r = -12 cos(theta) can be represented in rectangular form as x = -12 cos^2(theta) and y = -6sin(2theta).