The letters r and theta represent polar coordinates. Write each equation in rectangular coordinates (x, y) form.
Let t = theta
(1) r = sin(t) + 1
(2) r = sin(t) - cos(t)
Where you not given, or does your text not have formulas for changing from polar to rectangular????
I will do the first one
r^2 = x^2 + y^2 and sin(theta) = y/r
so you have
√(x^2+y^2) = y/√(x^2+y^2) + 1
multiply each term by √(x^2+y^2)
x^2+y^2 = y + √(x^2+y^2)
try the second one yourself, cos(t) = x/r
(3,2n/5)
To express the equations in rectangular coordinates (x, y) form, we need to use the relations between polar and rectangular coordinates.
For a point in polar coordinates (r, θ), the corresponding coordinates in rectangular form (x, y) are given by:
x = r * cos(θ)
y = r * sin(θ)
Let's apply these conversions to the given equations:
(1) r = sin(t) + 1:
Replacing r with its rectangular form:
x = (sin(t) + 1) * cos(t)
y = (sin(t) + 1) * sin(t)
Therefore, the equation in rectangular coordinates is:
x = (sin(t) + 1) * cos(t)
y = (sin(t) + 1) * sin(t)
(2) r = sin(t) - cos(t):
Replacing r with its rectangular form:
x = (sin(t) - cos(t)) * cos(t)
y = (sin(t) - cos(t)) * sin(t)
Therefore, the equation in rectangular coordinates is:
x = (sin(t) - cos(t)) * cos(t)
y = (sin(t) - cos(t)) * sin(t)
These equations express the given polar equations (1) and (2) in rectangular coordinates (x, y) form.