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.