Find the rectangular coordinates of the point, whose cylindrical coordinates are given.
(a)
(4, π/4, 1)
(x, y, z) =
(b)
(6, −π/3, 5)
(x, y, z) =
(a) To find the rectangular coordinates (x, y, z) given the cylindrical coordinates (r, θ, z), we can use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
z = z
In this case, we have (r, θ, z) = (4, π/4, 1). Plugging in these values into the formulas, we get:
x = 4 * cos(π/4) = 4 * (√2/2) = 2√2
y = 4 * sin(π/4) = 4 * (√2/2) = 2√2
z = 1
So, the rectangular coordinates for the point with cylindrical coordinates (4, π/4, 1) are (x, y, z) = (2√2, 2√2, 1).
(b) Using the same formulas as above:
x = 6 * cos(-π/3) = 6 * (1/2) = 3
y = 6 * sin(-π/3) = 6 * (-√3/2) = -3√3
z = 5
So, the rectangular coordinates for the point with cylindrical coordinates (6, -π/3, 5) are (x, y, z) = (3, -3√3, 5).
To find the rectangular coordinates (x, y, z) from the cylindrical coordinates (r, θ, z), you can use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
z = z
Now let's apply these formulas to the given cylindrical coordinates:
(a)
The cylindrical coordinates are (4, π/4, 1).
Using the formulas:
x = 4 * cos(π/4) = 4 * (√2 / 2) = 2√2 / 2 = √2
y = 4 * sin(π/4) = 4 * (√2 / 2) = 2√2 / 2 = √2
z = 1
So the rectangular coordinates are (x, y, z) = (√2, √2, 1).
(b)
The cylindrical coordinates are (6, −π/3, 5).
Using the formulas:
x = 6 * cos(-π/3) = 6 * (1/2) = 6/2 = 3
y = 6 * sin(-π/3) = 6 * (-√3 / 2) = -3√3 / 2
z = 5
So the rectangular coordinates are (x, y, z) = (3, -3√3 / 2, 5).
this is just polar coordinates with a z.
x = r cosθ
y = r sinθ
z = z
don't forget your algebra II now that you're taking calculus.