Write the equation with polar coordinates. Use "theta" for
θ
x+y=10
r=
r cos(theta) + r sin(theta) = 10
continuing on,
r(cosθ+sinθ) = 10
r * √2 sin(θ + π/4) = 10
r = 5√2 csc(θ + π/4)
Great work! Another possible final step could also be to solve for theta in terms of r:
sin(theta) + cos(theta) = 10/r
Dividing both sides by sqrt(2) and using the identity sin(π/4) = cos(π/4) = 1/sqrt(2), we get:
sin(theta + π/4) = 10/(r sqrt(2))
Taking the inverse sine of both sides, we have:
theta + π/4 = sin^(-1)(10/(r sqrt(2)))
Therefore:
theta = sin^(-1)(10/(r sqrt(2))) - π/4
This expression gives us theta in terms of r, which is another way to describe the polar equation of the line x + y = 10.
To convert the equation x + y = 10 into polar coordinates, we need to express x and y in terms of r and θ.
We can begin by expressing x and y in terms of r and θ using the following equations:
x = r cos(θ)
y = r sin(θ)
Now let's substitute these expressions back into the equation x + y = 10:
r cos(θ) + r sin(θ) = 10
This is the equation with polar coordinates.