How do you convert $(θ) = - 7 \frac{π}{4}$ $(theta)=-7pi/4$ to rectangular form?

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How do you convert $(θ) = - 7 \frac{π}{4}$ $(theta)=-7pi/4$ to rectangular form?

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Answer 1 · 2016-02-26T09:40:23

Equation in rectangular coordinates is $x + y = 0$ $x+y=0$

Explanation:

To convert equation $θ = - \frac{7 π}{4}$ $theta=−(7pi)/4$ , remember relation between polar and rectangular coordinates given by

$x = r cos θ$ $x=rcostheta$ and $y = r sin θ$ $y=rsintheta$ i.e. $r^{2} = x^{2} + y^{2}$ $r^2=x^2+y^2$ and $θ = {tan}^{- 1} (\frac{y}{x})$ $theta=tan^(-1)(y/x)$ or $tan θ = \frac{y}{x}$ $tantheta=y/x$

Hence the equation $θ = - \frac{7 π}{4}$ $theta=−(7pi)/4$ means

$tan (- \frac{7 π}{4}) = \frac{y}{x}$ $tan (-(7pi)/4)=y/x$ , but $tan (- \frac{7 π}{4}) = - 1$ $tan(-(7pi)/4)=-1$

Hence equation in rectangular coordinates is $\frac{y}{x} = - 1$ $y/x=-1$ i.e.

$x + y = 0$ $x+y=0$

Answer 2 · 2016-02-26T11:48:58

The radial line (half line ) y = x, in the first quadrant..

Explanation:

In polar coordinates, $θ$ $theta$ = constant represents a radial line and is a half-line.
The other half in the opposite direction is governed by $θ$ $theta$ = the constant + $π$ $pi$ .
Indeed, the given equation can be given directly as $θ$ $theta$ = $π$ $pi$ /4.

Answer 3 · 2016-02-26T13:21:34

one more Explanation

Explanation:

$y = r sin (- 7 \frac{π}{4})$ $y=rsin(-7pi/4)$
$\Rightarrow y = - r sin (7 \frac{π}{4})$ $=>y=-rsin(7pi/4)$ [since $sin (- θ) = - sin θ$ $sin(-theta)=-sintheta$ ]
$\Rightarrow y = - r sin (2 π - \frac{π}{4})$ $=>y=-rsin(2pi-pi/4)$
$\Rightarrow y = r sin (\frac{π}{4})$ $=>y=rsin(pi/4)$ [in 4th quadrant sin is negative]
$\Rightarrow y = \frac{r}{\sqrt{2}}$ $=>y=r/sqrt2$
again
$x = r cos (- 7 \frac{π}{4})$ $x=rcos(-7pi/4)$ [since $cos (- θ) = cos θ$ $cos(-theta)=costheta$ ]
$\Rightarrow x = r cos (7 \frac{π}{4})$ $=>x=rcos(7pi/4)$
$\Rightarrow x = r cos (2 π - \frac{π}{4})$ $=>x=rcos(2pi-pi/4)$
$\Rightarrow x = r cos (\frac{π}{4})$ $=>x=rcos(pi/4)$ [in 4th quadrant cos is positive]
$\Rightarrow x = \frac{r}{\sqrt{2}}$ $=>x=r/sqrt2$
hence
$x = y$ $x=y$

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How do you convert $(θ) = - 7 \frac{π}{4}$ $(theta)=-7pi/4$ to rectangular form?

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How do you convert $(θ) = - 7 \frac{π}{4}$ $(theta)=-7pi/4$ to rectangular form?