I) Express ( 2 , π 6 ) (2,\frac{\pi}{6}) ( 2 , 6 π ) in rectangular coordinates:
x = r c o s θ , x=rcos\theta, x = rcos θ ,
y = r s i n θ ; y=rsin\theta; y = rs in θ ;
r = 2 , θ = π 6 r=2,\space \theta=\frac{\pi}{6} r = 2 , θ = 6 π
x = 2 ∗ c o s ( π 6 ) , x=2*cos(\frac{\pi}{6}), x = 2 ∗ cos ( 6 π ) ,
y = 2 ∗ s i n ( π 6 ) . y=2*sin(\frac{\pi}{6}). y = 2 ∗ s in ( 6 π ) .
x = 2 ∗ 3 2 = 3 , x=2*\frac{\sqrt{3}}{2}=\sqrt{3}, x = 2 ∗ 2 3 = 3 ,
y = 2 ∗ 1 2 = 1. y=2*\frac{1}{2}=1. y = 2 ∗ 2 1 = 1.
Answer: x = 3 , y = 1. x=\sqrt{3},\space y=1. x = 3 , y = 1.
II) Express r = 1 + 2 c o s θ r=1+2cos\theta r = 1 + 2 cos θ from polar to cartesian form.
x = r c o s θ , x=rcos\theta, x = rcos θ ,
y = s i n θ . y=sin\theta. y = s in θ .
r = x 2 + y 2 r=\sqrt{x^2+y^2} r = x 2 + y 2
r = 1 + 2 c o s θ r=1+2cos\theta r = 1 + 2 cos θ
2 c o s θ = r − 1 2cos\theta=r-1 2 cos θ = r − 1
2 ( x r ) = r − 1 2(\frac{x}{r})=r-1 2 ( r x ) = r − 1
2 x = r 2 − r 2x=r^2-r 2 x = r 2 − r
2 x = ( x 2 + y 2 ) − x 2 + y 2 2x=(x^2+y^2)-\sqrt{x^2+y^2} 2 x = ( x 2 + y 2 ) − x 2 + y 2
Answer:
x 2 + y 2 = 2 x + x 2 + y 2 . x^2+y^2=2x+\sqrt{x^2+y^2}. x 2 + y 2 = 2 x + x 2 + y 2 .
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