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Chapter 6: Problem 54

Convert the polar equation to rectangular form. $$r=10$$

Short Answer

Expert verified
The rectangle form of the given polar equation is \( x^{2} + y^{2} = 100 \)

Step by step solution

01

Express x and y in terms of r

We can start by expressing x and y in terms of r. The coordinates for a polar system are \( (r, 胃) \), whereas for a Cartesian coordinate system, the coordinates are \( (x, y) \). The relationships between polar and rectangular coordinates are \( x = r \cos 胃 \) and \( y = r \sin 胃 \). Here, however, we don't have much to convert for this equation because it's a very simple polar equation.
02

Substitute the value of r

Now, in our specific equation, we have \( r = 10 \). This equals to the constant 10. So, there is no function of 胃 in this case. No matter the angle in polar coordinates, the radius will always be at distance 10 from the origin. This simply describes a circle with radius 10 in the polar coordinate system.
03

Translate to Rectangular Form

In order to convert it to rectangular form, the equation would be the equation of a circle of radius 10 centered at the origin. Draw on the relationships between the polar and rectangular systems, to translate from a circle of radius r in polar to a circle in rectangular, apply the Pythagorean theorem to the expressions for x and y: \( x^{2} + y^{2} = r^{2} \). Now, substituting the value of r from step 2 \( (=10) \) into this equation, gives \( x^{2} + y^{2} = 10^{2} = 100 \).
04

Conclusion

Consequently, \( x^{2} + y^{2} = 100 \) is the rectangular form of the given polar equation \( r = 10 \). This represents a circle centered at the origin (0, 0) with a radius of 10 units in the x-y plane.

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Most popular questions from this chapter

The graph of \(r=f(\theta)\) is rotated about the pole through an angle \(\phi\). Show that the equation of the rotated graph is \(r=f(\theta-\phi) .\)

\(r=4 \cos \theta\)

True or False? In Exercises 57 and 58, determine whether the statement is true or false. Justify your answer. In the polar coordinate system, if a graph that has symmetry with respect to the polar axis were folded on the line \(\theta=0\), the portion of the graph above the polar axis would coincide with the portion of the graph below the polar axis.

In Exercises 73-78, solve the trigonometric equation. $$ 6 \cos x-2=1 $$

In Exercises 73-78, solve the trigonometric equation. $$ 4 \sqrt{3} \tan \theta-3=1 $$
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