Question

Another representation in polar coordinates of the point (1 5 pi/6) is

Answer 1

hello :
note : the polar coordinates is : (r , θ)
Another representation is (x, y) : x=r cos(θ) and y = sin(θ) and : r =√(x²+y²)
in this exercice : r = 1 and θ = 5π/6
so : x= 1.cos(5π/6) and y = 1. sin(5π/6)
but : cos(5π/6) = cos (π - π/6) = - cos(π/6 ) = (- √3)/2
sin(5π/6) = sin (π - π/6) = sin(π/6 ) =1/2
Another representation is : ( (- √3)/2 , 1/2)

Answer 2

Answer: The representation of the point in Cartesian co-ordinate system is

`\left((\sqrt3)/(2),-(1)/(2)\right),~\left(-(\sqrt3)/(2),(1)/(2)\right).`

Step-by-step explanation: We are given to find the other representation of the point
`\left(1,(5\pi)/(6)\right)` in Cartesian system

We know that

if (x, y) are the co-ordinates of a point in two dimensional XY-plane and (r, Θ) are the co-ordinates of the point in polar co-ordinate system, then we have the following relations:

`x^2+y^2=r^2,~~~~~\tan \theta=(y)/(x).`

Given that

`(r,\theta)=\left(1,(5\pi)/(6)\right).`

Therefore, we have

`x^2+y^2=1^2\\\\\Rightarrow x^2+y^2=1,`

and

`\tan \theta=(y)/(x)\\\\\\\Rightarrow \tan (5\pi)/(6)=(y)/(x)\\\\\\\Rightarrow \tan(\pi-(\pi)/(6))=(y)/(x)\\\\\\\Rightarrow -\tan(\pi)/(6)=(y)/(x)\\\\\\\Rightarrow -(1)/(\sqrt3)=(y)/(x)\\\\\\\Rightarrow x=-\sqrt3y.`

So,

`x^2+y^2=1\\\\\Rightarrow (-\sqrt3y)^2+y^2=1\\\\\Rightarrow 4y^2=1\\\\\Rightarrow y^2=(1)/(4)\\\\\\\Rightarrow y=\pm(1)/(2).`

And,

When
`x=(1)/(2),`,

`-(1)/(\sqrt3)=(1)/(2x)\\\\\Rightarrow x=-(\sqrt3)/(2).`

When
`x=-(1)/(2),`,

`-(1)/(\sqrt3)=-(1)/(2x)\\\\\Rightarrow x=(\sqrt3)/(2).`

Thus, the representation of the point in Cartesian co-ordinate system is

`\left((\sqrt3)/(2),-(1)/(2)\right),~\left(-(\sqrt3)/(2),(1)/(2)\right).`