# Find all points of intersection of the given curves. (Assume

0 ≤ θ ≤ 2π.

Order your answers from smallest to largest θ. If an intersection occurs at the pole, enter POLE in the first answer blank.)

r = 1 − cos θ, r = 1 + sin θ

## well, just do

1 - cosθ = 1 + sinθ

sinθ = -cosθ

sin π/4 = cos π/4

So, in QII and QIV,

sin 3π/4 = -cos 3π/4

sin 7π/4 = -cos 7π/4

## To find all points of intersection between the curves, we need to set the equations equal to each other and solve for θ.

1 - cos θ = 1 + sin θ

Rearranging the equation:

sin θ + cos θ = 0

Using the Pythagorean identity sin²θ + cos²θ = 1, we can rewrite the equation as:

sin θ + cos θ = √2 * sin(θ + π/4)

Setting the equation equal to zero:

√2 * sin(θ + π/4) = 0

sin(θ + π/4) = 0

θ + π/4 = nπ

θ = -π/4 + nπ, where n is an integer.

Since the values of θ should be between 0 and 2π, we can find the intersection points as follows:

θ = -π/4 + 0π = -π/4

θ = -π/4 + 1π = 3π/4

θ = -π/4 + 2π = 7π/4

So, the points of intersection are (-π/4, 0), (3π/4, 0), and (7π/4, 0).

## To find the points of intersection between the two curves, we need to equate the values of r for both curves and solve for θ.

The curves are represented by the following equations:

Curve 1: r = 1 - cos(θ)

Curve 2: r = 1 + sin(θ)

To find the points of intersection:

1. Set the values of r in both curves equal to each other:

1 - cos(θ) = 1 + sin(θ)

2. Rearrange the equation to isolate sin(θ):

sin(θ) = cos(θ) - 1

3. Square both sides of the equation to eliminate the square root:

sin^2(θ) = cos^2(θ) - 2cos(θ) + 1

4. Use the trigonometric identity sin^2(θ) + cos^2(θ) = 1 to simplify the equation:

1 - cos^2(θ) = cos^2(θ) - 2cos(θ) + 1

5. Rearrange the equation and simplify:

2cos^2(θ) - 2cos(θ) = 0

6. Factor out 2cos(θ):

2cos(θ)(cos(θ) - 1) = 0

7. Set each factor equal to zero and solve for θ:

a) cos(θ) = 0

θ = π/2, 3π/2 (since 0 ≤ θ ≤ 2π)

b) cos(θ) - 1 = 0

cos(θ) = 1

θ = 0, 2π (since 0 ≤ θ ≤ 2π)

Therefore, the possible values of θ where the two curves intersect are:

θ = 0, π/2, 3π/2, 2π

Note: We ordered the values of θ from smallest to largest. Since θ = 0 and θ = 2π represent the same point at the pole, you may enter "POLE" for the first answer blank. Otherwise, you would enter θ = 0 for the first answer blank and continue with the other values of θ in increasing order.