Chapter 13: Q 36 (page 1066)

Describe the graphs of the equations and provide alternative equations in the specified coordinate systems.
Change $ϕ = \frac{π}{4}$ to the rectangular and spherical systems.

Short Answer

Expert verified

The equation in rectangular coordinates is $x^{2} + y^{2} - z^{2} = 0$

Step by step solution

Step 1:Given information

The given expression is $ϕ = \frac{π}{4}$

Simplification (a)

To convert $ϕ = \frac{π}{4}$ into rectangular coordinates

We have

$r = \sqrt{x^{2} + y^{2} + z^{2}}, \tan θ = \frac{y}{x}, \cos ϕ = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}$

$\Rightarrow \cos ϕ = \frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}$

$\Rightarrow ϕ = \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$

Substituting $\Rightarrow ϕ = \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$ in $ϕ = \frac{π}{4}$

$\Rightarrow \frac{π}{4} = \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$

$\Rightarrow \cos \frac{π}{4} = (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$

$\Rightarrow (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}) = \frac{1}{\sqrt{2}}$

$\Rightarrow (\frac{z^{2}}{x^{2} + y^{2} + z^{2}}) = \frac{1}{2}$

$\Rightarrow x^{2} + y^{2} + z^{2} = 2 z^{2}$

$\Rightarrow x^{2} + y^{2} - z^{2} = 0$

Hence,The equation in rectangular coordinates is $x^{2} + y^{2} - z^{2} = 0$

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Describe the graphs of the equations and provide alternative equations in the specified coordinate systems.
Change $ϕ = \frac{π}{4}$ to the rectangular and spherical systems.

Short Answer

Step by step solution

Step 1:Given information

Simplification (a)

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Describe the graphs of the equations and provide alternative equations in the specified coordinate systems.Change ϕ=π4to the rectangular and spherical systems.

Short Answer

Step by step solution

Step 1:Given information

Simplification (a)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Calculus

Discrete Mathematics

Statistics

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

Describe the graphs of the equations and provide alternative equations in the specified coordinate systems.
Change $ϕ = \frac{π}{4}$ to the rectangular and spherical systems.