{
"query": {
"display": "polar to cartesian $$\\left(2,\\:\\frac{3π}{2}\\right)$$",
"symbolab_question": "POLAR#cartesian (2,\\frac{3π}{2})"
},
"solution": {
"level": "PERFORMED",
"subject": "Pre Calculus",
"topic": "Polar Coordinates",
"subTopic": "Cartesian",
"default": "(0,-2)"
},
"steps": {
"type": "interim",
"title": "Convert $$\\left(2,\\:\\frac{3π}{2}\\right)\\:$$to Cartesian coordinates:$${\\quad}\\left(0,\\:-2\\right)$$",
"steps": [
{
"type": "definition",
"title": "Definition",
"text": "To convert Polar coordinates $$\\left(r,\\:\\theta\\right)\\:$$to Cartesian coordinates $$\\left(x,\\:y\\right)\\:$$apply:<br/>$$x=r\\cdot\\cos\\left(\\theta\\right)\\quad\\:y=r\\cdot\\sin\\left(\\theta\\right)$$",
"secondary": [
"$$r=2$$",
"$$θ=\\frac{3π}{2}$$"
]
},
{
"type": "step",
"primary": "$$x=r\\cdot\\cos\\left(\\theta\\right)$$",
"result": "x=2\\cos\\left(\\frac{3π}{2}\\right)"
},
{
"type": "interim",
"title": "$$2\\cos\\left(\\frac{3π}{2}\\right)=0$$",
"input": "2\\cos\\left(\\frac{3π}{2}\\right)",
"steps": [
{
"type": "interim",
"title": "$$\\cos\\left(\\frac{3π}{2}\\right)=0$$",
"input": "\\cos\\left(\\frac{3π}{2}\\right)",
"result": "=2\\cdot\\:0",
"steps": [
{
"type": "interim",
"title": "Rewrite using trig identities:$${\\quad}\\cos\\left(π\\right)\\cos\\left(\\frac{π}{2}\\right)-\\sin\\left(π\\right)\\sin\\left(\\frac{π}{2}\\right)$$",
"input": "\\cos\\left(\\frac{3π}{2}\\right)",
"result": "=\\cos\\left(π\\right)\\cos\\left(\\frac{π}{2}\\right)-\\sin\\left(π\\right)\\sin\\left(\\frac{π}{2}\\right)",
"steps": [
{
"type": "step",
"primary": "Write $$\\cos\\left(\\frac{3π}{2}\\right)\\:$$as $$\\cos\\left(π+\\frac{π}{2}\\right)$$",
"result": "=\\cos\\left(π+\\frac{π}{2}\\right)"
},
{
"type": "step",
"primary": "Use the Angle Sum identity: $$\\cos\\left(s+t\\right)=\\cos\\left(s\\right)\\cos\\left(t\\right)-\\sin\\left(s\\right)\\sin\\left(t\\right)$$",
"result": "=\\cos\\left(π\\right)\\cos\\left(\\frac{π}{2}\\right)-\\sin\\left(π\\right)\\sin\\left(\\frac{π}{2}\\right)"
}
],
"meta": {
"interimType": "Trig Rewrite Using Trig identities Title 0Eq"
}
},
{
"type": "interim",
"title": "Use the following trivial identity:$${\\quad}\\cos\\left(π\\right)=\\left(-1\\right)$$",
"input": "\\cos\\left(π\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=\\left(-1\\right)"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "interim",
"title": "Use the following trivial identity:$${\\quad}\\cos\\left(\\frac{π}{2}\\right)=0$$",
"input": "\\cos\\left(\\frac{π}{2}\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=0"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "interim",
"title": "Use the following trivial identity:$${\\quad}\\sin\\left(π\\right)=0$$",
"input": "\\sin\\left(π\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=0"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "interim",
"title": "Use the following trivial identity:$${\\quad}\\sin\\left(\\frac{π}{2}\\right)=1$$",
"input": "\\sin\\left(\\frac{π}{2}\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=1"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "step",
"result": "=\\left(-1\\right)\\cdot\\:0-0\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=0"
}
],
"meta": {
"solvingClass": "Trig Evaluate",
"interimType": "Trig Evaluate"
}
},
{
"type": "step",
"primary": "Apply rule $$0\\cdot\\:a=0$$",
"result": "=0"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7M3U1zttpUVfifMFzC3pOLJgZZwFPaLazbEUSn25iPmR1g99dC9fj9sg0EHzBIRDRd79UrkSVT0SCLs80Lgihlw/dmvsYNaLIHPMc9wnCrm76SUFivy/FGl6NB0I4XEdAJLd1ohke2Wgml78++2zI0g=="
}
},
{
"type": "step",
"result": "x=0"
},
{
"type": "step",
"primary": "$$y=r\\cdot\\sin\\left(\\theta\\right)$$",
"result": "y=2\\sin\\left(\\frac{3π}{2}\\right)"
},
{
"type": "interim",
"title": "$$2\\sin\\left(\\frac{3π}{2}\\right)=-2$$",
"input": "2\\sin\\left(\\frac{3π}{2}\\right)",
"steps": [
{
"type": "interim",
"title": "$$\\sin\\left(\\frac{3π}{2}\\right)=-1$$",
"input": "\\sin\\left(\\frac{3π}{2}\\right)",
"result": "=2\\left(-1\\right)",
"steps": [
{
"type": "interim",
"title": "Rewrite using trig identities:$${\\quad}\\sin\\left(π\\right)\\cos\\left(\\frac{π}{2}\\right)+\\cos\\left(π\\right)\\sin\\left(\\frac{π}{2}\\right)$$",
"input": "\\sin\\left(\\frac{3π}{2}\\right)",
"result": "=\\sin\\left(π\\right)\\cos\\left(\\frac{π}{2}\\right)+\\cos\\left(π\\right)\\sin\\left(\\frac{π}{2}\\right)",
"steps": [
{
"type": "step",
"primary": "Write $$\\sin\\left(\\frac{3π}{2}\\right)\\:$$as $$\\sin\\left(π+\\frac{π}{2}\\right)$$",
"result": "=\\sin\\left(π+\\frac{π}{2}\\right)"
},
{
"type": "step",
"primary": "Use the Angle Sum identity: $$\\sin\\left(s+t\\right)=\\sin\\left(s\\right)\\cos\\left(t\\right)+\\cos\\left(s\\right)\\sin\\left(t\\right)$$",
"result": "=\\sin\\left(π\\right)\\cos\\left(\\frac{π}{2}\\right)+\\cos\\left(π\\right)\\sin\\left(\\frac{π}{2}\\right)"
}
],
"meta": {
"interimType": "Trig Rewrite Using Trig identities Title 0Eq"
}
},
{
"type": "interim",
"title": "Use the following trivial identity:$${\\quad}\\sin\\left(π\\right)=0$$",
"input": "\\sin\\left(π\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=0"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "interim",
"title": "Use the following trivial identity:$${\\quad}\\cos\\left(\\frac{π}{2}\\right)=0$$",
"input": "\\cos\\left(\\frac{π}{2}\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=0"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "interim",
"title": "Use the following trivial identity:$${\\quad}\\cos\\left(π\\right)=\\left(-1\\right)$$",
"input": "\\cos\\left(π\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=\\left(-1\\right)"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "interim",
"title": "Use the following trivial identity:$${\\quad}\\sin\\left(\\frac{π}{2}\\right)=1$$",
"input": "\\sin\\left(\\frac{π}{2}\\right)",
"steps": [
{
"type": "step",
"primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$"
},
{
"type": "step",
"result": "=1"
}
],
"meta": {
"interimType": "Trig Trivial Angle Value Title 0Eq"
}
},
{
"type": "step",
"result": "=0\\cdot\\:0+\\left(-1\\right)\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=-1"
}
],
"meta": {
"solvingClass": "Trig Evaluate",
"interimType": "Trig Evaluate"
}
},
{
"type": "interim",
"title": "Simplify",
"input": "2\\left(-1\\right)",
"steps": [
{
"type": "step",
"primary": "Remove parentheses: $$\\left(-a\\right)=-a$$",
"result": "=-2\\cdot\\:1"
},
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:1=2$$",
"result": "=-2"
}
],
"meta": {
"interimType": "Generic Simplify 0Eq"
}
},
{
"type": "step",
"result": "=-2"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver"
}
},
{
"type": "step",
"result": "y=-2"
},
{
"type": "step",
"primary": "The Cartesian coordinates of $$\\left(2,\\:\\frac{3π}{2}\\right)$$",
"result": "\\left(0,\\:-2\\right)"
}
]
}
}
Solution
polar to cartesian
Solution
Solution steps
The Cartesian coordinates of