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 write the rectangular equation (x – 4)2 + y2 = 16 in polar form, we can convert it to the polar variables. In polar coordinates, we have x = r cos(theta) and y = r sin(theta). Substituting these values into the equation, we get:

(r cos(theta) – 4)2 + (r sin(theta))^2 = 16

Expanding and simplifying the equation yields:

r^2 cos^2(theta) – 8r cos(theta) + 16 + r^2 sin^2(theta) = 16

Using the trigonometric identity cos^2(theta) + sin^2(theta) = 1, we can simplify further:

r^2 - 8r cos(theta) + 16 = 16

Finally, subtracting 16 from both sides gives us the polar form of the equation:

r^2 - 8r cos(theta) = 0

2. To write the equation –2x + 6y = 7 in polar form, we need to express x and y in terms of r and theta. Using the polar coordinate conversions x = r cos(theta) and y = r sin(theta), we can rewrite the given formula as:

–2(r cos(theta)) + 6(r sin(theta)) = 7

Simplifying this equation further, we get:

–2r cos(theta) + 6r sin(theta) = 7

Next, we divide both sides by r to isolate the trigonometric terms:

–2 cos(theta) + 6 sin(theta) = 7/r

Finally, rewriting the trigonometric terms in terms of a single trigonometric function, we obtain the polar form of the equation:

r(-2 cos(theta) + 6 sin(theta)) = 7

3. To graph the polar equation 2 = r cos(theta – 20°), we can first rewrite it in terms of r and theta. We need to isolate r by dividing both sides of the equation by cos(theta – 20°):

2 / cos(theta – 20°) = r

Now, we can plot the graph. For each value of theta, calculate the corresponding value of r using the equation r = 2 / cos(theta – 20°). Plot the points (r, theta), and connect them to observe the resulting shape.

4. To write the polar equation r = –12 cos(theta) in rectangular form, we can use the polar coordinate conversions x = r cos(theta) and y = r sin(theta). Substituting these values, we get:

x = –12 cos(theta) cos(theta)
y = –12 cos(theta) sin(theta)

Next, simplifying, we obtain:

x = –12 cos^2(theta)
y = –12 cos(theta) sin(theta)

Finally, using the trigonometric identity sin(2theta) = 2 sin(theta) cos(theta), we can rewrite y in terms of x:

y = –6 sin(2theta)

So, the rectangular form of the polar equation is x = –12 cos^2(theta) and y = –6 sin(2theta).