Can someone please help me with converting polar equations to rectangular form? My life depends on it.?

Question

My math teacher she gave the class an overview of the material, but did not provide enough examples. Since, this stuff is so new and strange to me I need help on on how to do the following five questions.

Convert the equation from polar to rectagular form.

1. ø = pi /6

2. r = 4

3. r = -3csc "fish symbol"

4. r^2 = cos ø

5. r = 6/(2-3 sin ø)

Thank you so much to anyone who is willing to help me. Help of any or all of the problems is appreciated.

Answer 1

1. ø = pi /6

ø= arctan (y/x)
arctan(y/x) = pi/6
y/x = tan(pi/6)
y/x =0.577
y= 0.577 x

2. r = 4
r=sqrt(x^2+y^2)
so sqrt(x^2+y^2) = 4
x^2+y^2 = 16

3. r = -3csc "fish symbol"

4. r^2 = cos ø
x^2 + y^2 = x/(sqrt(x^2+y^2)
sqrt(x^2+y^2) (x^2+y^2) =x
so
(x^2+y^2)^{3/2} =x

5. r = 6/(2-3 sin ø)
sqrt(x^2+y^2) = 6/(2-3 y/sqrt(x^2+y^2) )
sqrt(x^2+y^2) = 6/ [( 2sqrt(x^2+y^2) -3y )/ sqrt(x^2+y^2)]
sqrt(x^2+y^2) = 6sqrt(x^2+y^2) / ( 2sqrt(x^2+y^2) -3y )
1= 6/( 2sqrt(x^2+y^2) -3y )
2sqrt(x^2+y^2) -3y =6
2sqrt(x^2+y^2) = 6+3y
sqrt(x^2+y^2) = 3 + 3y/2
x^2+y^2 = (3+ 3y/2)^2

Answer 2

1. ø = pi /6

tan ø = y/x
y = x tan ø = x tan(pi/6) = x sqrt(3) = 0.577 x


2. r = 4

r = sqrt(x^2 + y^2) = 4
x^2 + y^2 = 16


3. r = -3 csc ø "fish symbol"
I added the "ø" to the cosecant. It is needed.

r = -3 csc ø = -3 / sin ø
sqrt(x^2 + y^2) = -3 / [y / sqrt(x^2 + y^2)]
sqrt(x^2 + y^2) = -3 sqrt(x^2 + y^2) / y
1 = -3 / y
y = -3


4. r^2 = cos ø

x^2 + y^2 = x / sqrt(x^2 + y^2)
(x^2 + y^2)^(3/2) = x


5. r = 6 / (2 - 3 sin ø)

sqrt(x^2+y^2) = 6 / (2 - 3y / sqrt(x^2+y^2))
sqrt(x^2+y^2) = 6 sqrt(x^2+y^2) / [(2 sqrt(x^2+y^2) - 3y]
1 = 6 / (2 sqrt(x^2+y^2) - 3y)
2 sqrt(x^2+y^2) - 3y = 6
2 sqrt(x^2+y^2) = 3y + 6 = 3 (y + 2)
4x^2 + 4y^2 = 9 (y + 2)^2 = 9(y^2 + 4y + 4)
4x^2 + 4y^2 = 9y^2 + 36y + 36
4x^2 - 5y^2 - 36y = 36
4x^2 - 5[y^2 - (36/5)y + (18/5)^2] = 36 + 5(18/5)^2
4x^2 - 5[y - (18/5)]^2 = 100.8
4x^2 - 5(y - 3.6)^2 = 100.8

Answer 3

You're gonna die.