Convert r = 1 / (6cos(θ)+8sin(θ)) to an equation in rectangular coordinates (i.e., in terms of x and y).
A similar example I found, where r = 1 / (cos (θ) − sin (θ)), found the rectangular equation to be y = x - 1. How would this answer change for my problem now that sin and cos have coefficients?
the coefficients just change the slope of the line.
r(6cosθ+8sinθ) = 1
6x+8y = 1
To convert the polar equation r = 1 / (6cos(θ)+8sin(θ)) to rectangular coordinates, we can use the following trigonometric identities:
cos(θ) = x / r
sin(θ) = y / r
Substituting these identities into the polar equation, we get:
r = 1 / (6(x / r) + 8(y / r))
Multiplying both sides by r, we can simplify it further:
r^2 = 1 / (6x + 8y)
Now, to express this equation in terms of x and y, we need to replace r^2 with x^2 + y^2:
x^2 + y^2 = 1 / (6x + 8y)
This is the equation of the curve in rectangular coordinates. It represents an implicit relationship between x and y values.
For the similar example you mentioned (r = 1 / (cos(θ) - sin(θ))), the final equation was y = x - 1. In this specific case, the coefficients of sin(θ) and cos(θ) were 1, and they equaled 1 in the denominator.
In your given problem, the coefficients of sin(θ) and cos(θ) are 8 and 6, respectively. Thus, the equation in rectangular coordinates is not a linear function like y = x - 1. It is an implicitly defined curve equation that relates x and y through the equation x^2 + y^2 = 1 / (6x + 8y).