A curve with polar equation
r=(8)/7sinx + 36cosx
represents a line. This line has a Cartesian equation of the form
y=mx+b ,where m and b are constants. Give the formula
for y in terms of x . For example, if the line had equation
y=2x+3 then the answer would be 2x+3 .
you must mean r = 8/(7sinØ + 36cosØ)
It would be very confusing to have x as an angle and then use it in the linear function
Also I tried your equation the way you typed it and got nowhere.
conversion: r^2 = x^2 + y^2, and sinØ = y/r, cosØ = x/r
r = 8/((7y/x) + 36x/r))
r = 8r/(7y + 36x)
1 = 8/(7y + 36x)
36x + 7y = 8 is the linear equation
How do you convert 36x + 7y = 8 into y=mx+b format?
that is the easy part ....
36x + 7y = 8
7y = -36x + 8
y = (-36/7)x + 8/7
To find the Cartesian equation of the line represented by the polar equation, we can use the following relationships:
x = r * cos(x)
y = r * sin(x)
Given the polar equation: r = (8/7)sin(x) + 36cos(x)
We can substitute these equations into the polar equation:
x = ((8/7)sin(x) + 36cos(x)) * cos(x)
y = ((8/7)sin(x) + 36cos(x)) * sin(x)
Simplifying these equations, we get:
x = (8/7)sin(x)cos(x) + 36cos^2(x)
y = (8/7)sin^2(x) + 36sin(x)cos(x)
Now, we can express y in terms of x. By dividing the second equation by the first equation, we can eliminate the trigonometric functions:
y/x = ((8/7)sin^2(x) + 36sin(x)cos(x)) / ((8/7)sin(x)cos(x) + 36cos^2(x))
Next, let's simplify the equation:
y/x = (8/7)sin(x) + 36cos(x) / (8/7)cos(x) + 36cos^2(x)/cos(x)
y/x = (8sin(x) + 36cos(x)) / (8cos(x) + 36cos^2(x)/cos(x))
Simplifying further, we get:
y/x = (8sin(x) + 36cos(x)) / (8cos(x) + 36cos(x))
y/x = (8sin(x) + 36cos(x)) / (44cos(x))
Now, multiplying both sides by x, we have:
y = (8sin(x) + 36cos(x)) / 44cos(x) * x
Simplifying the equation, we obtain the Cartesian equation of the line:
y = (2sin(x) + 9cos(x)) / 11cos(x) * x
Therefore, the equation of the line in the form y = mx + b is:
y = (2sin(x) + 9cos(x)) / 11cos(x) * x