Answered: (-4, 4V3)

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter4: Polynomials

Section4.2: Adding And Subtracting Polynomials

Problem 17OE

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Question

Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.

Expert Solution

Step 1

the given point in cartesian coordinate system is:

$(- 4, 4 \sqrt{3})$

Step 2

as we know that

$x = r \cos θ y = r \sin θ$

therefore,

$\begin{array}{rcl} - 4 & = & r \cos θ (1) \\ 4 \sqrt{3} & = & r \sin θ (2) \end{array}$

dividing equation (2) by (1).

$\frac{r \sin θ}{r \cos θ} = \frac{4 \sqrt{3}}{- 4} \tan θ = - \sqrt{3} \tan θ = - \tan (\frac{π}{3}) \tan θ = \tan (\frac{- π}{3})$

Step 3

we know that if tanx=tany then

$x = n π + y$

therefore,

if $\tan θ = \tan (\frac{- π}{3})$

then $\begin{array}{rcl} θ & = & n π + (\frac{- π}{3}) \\ θ & = & nπ - \frac{π}{3} \end{array}$

put n=1

$\begin{array}{rcl} θ & = & π - \frac{π}{3} \\ = & \frac{2 π}{3} \end{array}$

if $θ = \frac{2 π}{3}$

then $\sin θ > 0 \cos θ < 0$

and we want sin $θ$ to be positive and cos $θ$ to be negative.

so consider this value of $θ$ so while calculating the value of $θ$ consider also this fact that we want sin $θ$ to be positive and cos $θ$ to be negative.

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