Jerold

2020-11-22

To find: The equivalent polar equation for the given rectangular-coordinate equation.

Given:

$\text{}x=\text{}r\mathrm{cos}\theta$

$\text{}y=\text{}r\mathrm{sin}\theta$

b. From rectangular to polar:

$r=\pm \sqrt{{x}^{2}\text{}+\text{}{y}^{2}}$

$\mathrm{cos}\theta =\frac{x}{r},\mathrm{sin}\theta =\frac{y}{r},\mathrm{tan}\theta =\frac{x}{y}$

Calculation:

Given: equation in rectangular-coordinate is$y=x$ .

Converting into equivalent polar equation -

$y=x$

Put$x=r\mathrm{cos}\theta ,\text{}y=r\mathrm{sin}\theta ,$

$\Rightarrow \text{}r\mathrm{sin}\theta =r\mathrm{cos}\theta$

$\Rightarrow \text{}\frac{\mathrm{sin}\theta}{\mathrm{cos}\theta}=1$

$\Rightarrow \text{}\mathrm{tan}\theta =1$

Thus, desired equivalent polar equation would be$\theta =1$

Given:

b. From rectangular to polar:

Calculation:

Given: equation in rectangular-coordinate is

Converting into equivalent polar equation -

Put

Thus, desired equivalent polar equation would be

Arnold Odonnell

Skilled2020-11-23Added 109 answers

Concept used:

Conversion formulafor coordinate systems are given as -

a. From polar to rectangular:

$\text{}x=\text{}r\mathrm{cos}\theta$

$\text{}y=\text{}r\mathrm{sin}\theta$

b. From rectangular to polar:

$r=\pm \sqrt{{x}^{2}\text{}+\text{}{y}^{2}}$

$\mathrm{cos}\theta =\frac{x}{r},\mathrm{sin}\theta =\frac{y}{r},\mathrm{tan}\theta =\frac{x}{y}$

Calculation:

Given: equation in rectangular-coordinate is$y=x$ .

Converting into equivalent polar equation -

$y=x$

Put$x=r\mathrm{cos}\theta ,\text{}y=r\mathrm{sin}\theta$

$\Rightarrow \text{}r\mathrm{sin}\theta =r\mathrm{cos}\theta$

$\Rightarrow \text{}\frac{\mathrm{sin}\theta}{\mathrm{cos}\theta}=1$

$\Rightarrow \text{}\mathrm{tan}\theta =1$

Thus, desired equivalent polar equation would be$\theta =1$

Conversion formulafor coordinate systems are given as -

a. From polar to rectangular:

b. From rectangular to polar:

Calculation:

Given: equation in rectangular-coordinate is

Converting into equivalent polar equation -

Put

Thus, desired equivalent polar equation would be

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