Converting polar equations to rectangular form

Question

Write r=5/(3cos(θ)+8sin(θ)) in rectangular form

A step by step explanation would be greatly appreciated, Thanks in advance!

Answer 1

1.) Look up governing equations

They are [numbers in brackets bookmark the equation]:

x = rcos(θ) [1]

y = rsin(θ) [2]

x² + y² = r² [3]

θ = tan(y/x) [4]

With these we can now advance

2.) Simplify equation to make it more manageable:

a.) Multiply each side by the denominator on the right hand side, the equation would then read:

r[3cos(θ) + 8sin(θ)] = 5

b.) Note the fact that the 'r' can be distributed:

3r cos(θ) + 8r sin(θ) = 5

c.) Use governing equations to substitute in the correct form:

Let's break this down into parts, we'll inspect the "3r cos(θ)" first:

Use equation [1] since it uses the same trig function:

x = rcos(θ) implies 3x = 3rcos(θ) (if you divide out the 3, you'll see that they are equal and thus the equation proves that constant multiples are just kept in the process of converting)

So 3r cos(θ) = 3x

Now we'll inspect "8r cos(θ)":

Same idea applies but a new trig function

So 8r sin(θ) = 8y by equation [2]

3.) Reevaluate the new substituted equation:

3x + 8y = 5

Solve for 'y' like earlier math courses:

y = (5 - 3x) / 8