Please help me any one of these please! Thank You to any one
r=2-6 cos(θ) convert this polar coordinate equation into an equation in rectangular coordinates(x,y)
given (-8,8) convert this point into polar coordinates with 6 >= 0 and 0<=θ<= 2pi
given x^2+ (y-5)^2=25 cover this rectangular coordinate equation in to an equation in polar coordinates (r,θ)
To convert the polar coordinate equation r = 2 - 6cos(θ) into an equation in rectangular coordinates (x, y), we can use the following relationships:
x = rcos(θ)
y = rsin(θ)
Substituting these equations into the given polar equation, we get:
x = (2 - 6cos(θ))cos(θ)
y = (2 - 6cos(θ))sin(θ)
Therefore, the equation in rectangular coordinates is:
x = 2cos(θ) - 6cos^2(θ)
y = 2sin(θ) - 6cos(θ)sin(θ)
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To convert the point (-8, 8) into polar coordinates (r, θ), we can use the following relationships:
r = sqrt(x^2 + y^2)
θ = arctan(y/x)
Substituting the given values, we have:
r = sqrt((-8)^2 + 8^2) = sqrt(64 + 64) = sqrt(128) = 8√2
θ = arctan(8/(-8)) = arctan(-1) = -π/4 (when we choose 6 >= 0 and 0 <= θ <= 2π, we can consider arctan(-1) as -π/4)
Therefore, the polar coordinates of the point (-8, 8) are (8√2, -π/4).
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To convert the rectangular coordinate equation x^2 + (y - 5)^2 = 25 into an equation in polar coordinates (r, θ), we can use the following relationships:
x = rcos(θ)
y = rsin(θ)
Substituting these equations into the given rectangular equation, we get:
(rcos(θ))^2 + (rsin(θ) - 5)^2 = 25
Expanding and simplifying, we have:
r^2cos^2(θ) + r^2sin^2(θ) - 10rsin(θ) + 25 = 25
Simplifying further:
r^2(cos^2(θ) + sin^2(θ)) - 10rsin(θ) = 0
Since cos^2(θ) + sin^2(θ) = 1, the equation becomes:
r^2 - 10rsin(θ) = 0
This is the equation in polar coordinates (r, θ) corresponding to the given rectangular coordinate equation x^2 + (y - 5)^2 = 25.