Question

Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. (a) (5, 3π/2) (x, y) = (b) (3 2 , π/4) (x, y) = (c) (−1, −π/6) (x, y)

Answer 1

Answer:

$A)(5,\frac{3\pi}{2})=(0,-5)\:B)(\frac{3}{2} , \frac{\pi}{4} ) \Rightarrow (\frac{3\sqrt{2}}{4} ,\frac{3\sqrt{2}}{4} )\:C)(-1,\frac{\pi}{6})= (\frac{\sqrt{3}}{2},\frac{1}{2})$

Step-by-step explanation:

1) Check the Polar Coordinates on the graph.

2) A polar coordinate is given in $(r,\theta)$ Let's transform each polar coordinate into Cartesian one. Considering:

A)

$x=rcos\theta\\y=rsin\theta$

$A) (5,\frac{3\pi}{2})\\cos(\theta)=\frac{x}{5}\\cos(\frac{3\pi}{2})=\frac{x}{5}\:Cross\: Multiplying\\x=5 cos(\frac{3\pi}{2})\\x =5 cos(0)\:x=0\\sin(\frac{3\pi}{2})=\frac{y}{5}\\y=5sin(-1)\\y=-5\\(0,-5)$

B) (b) (3/2 , π/4)

$B) (\frac{3}{2},\frac{\pi}{4})\\cos(\theta)=\frac{x}{\frac{3}{2}}\\\frac{3}{2}cos(\frac{\pi}{4})=x\:Cross\: Multiplying\\x=\frac{3}{2}*\frac{\sqrt{2}}{2}\\x =\frac{3\sqrt{2}}{4} \approx 1.06$

$y=\frac{3}{2}sin(\frac{\pi}{4})\Rightarrow y =\frac{3\sqrt{2}}{4}$

$B)(\frac{3}{2} , \frac{\pi}{4} ) \Rightarrow (\frac{3\sqrt{2}}{4} ,\frac{3\sqrt{2}}{4} )$

C) (c) (−1, −π/6)

$C) (-1,-\frac{\pi}{6})\\cos(\theta)=\frac{x}{-1}\\-cos(\frac{-\pi}{6})=x\:Cross\: Multiplying\\x=\frac{\sqrt{3}}{2}$

$(-1,-\frac{\pi}{6})\\sin(\theta)=\frac{x}{-1}\\-sin(\frac{-\pi}{6})=x\:Cross\: Multiplying\\x=\frac{1}{2}$

$C)(-1,\frac{\pi}{6})= (\frac{\sqrt{3}}{2},\frac{1}{2})$