Chapter 30: Chapter 30

Problem 975

Convert the equation \(\mathrm{r}=\tan \theta+\cot \theta\) to an equation in cartesian coordinates.

Short Answer

Expert verified

The short answer is: The given polar equation \(\mathrm{r}=\tan \theta+\cot \theta\) can be converted to the Cartesian coordinates equation \((x^2 + y^2)^{3/2} = xy\).

See the step by step solution

Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Rewrite the equation in terms of sine and cosine functions

The given equation is: \(\mathrm{r}=\tan \theta+\cot \theta\) Since we know that: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), The equation becomes: \(\mathrm{r}=\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}\)

Step 2: Find a common denominator and simplify the equation

To simplify the equation, we will find a common denominator for the fractions, which is \(\cos \theta \sin \theta\). The equation now becomes: \(\mathrm{r} = \frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta \sin \theta}\) As we know, \(\sin^2 \theta + \cos^2 \theta = 1\), so the equation simplifies to: \(\mathrm{r} = \frac{1}{\cos \theta \sin \theta}\)

Step 3: Substitute the polar coordinates with Cartesian coordinates

Now, we will use the polar to Cartesian coordinate transformations: \(x = r \cos \theta\) \(y = r \sin \theta\) We already have the equation: \(\mathrm{r} = \frac{1}{\cos \theta \sin \theta}\) Substitute \(\cos \theta\) with \(\frac{x}{r}\) and \(\sin \theta\) with \(\frac{y}{r}\): \(\mathrm{r} = \frac{1}{\frac{x}{r} \cdot \frac{y}{r}}\)

Step 4: Simplify the equation and solve for \(r\)

Now, simplify the equation: \(\mathrm{r} = \frac{1}{\frac{xy}{r^2}}\) \(\mathrm{r} = \frac{r^2}{xy}\) Now, multiply both sides by \(xy\): \(r^3 = x * y\)

Step 5: Substitute back Cartesian coordinates for \(r\)

Finally, replace \(r\) with \(\sqrt{x^2 + y^2}\) (since \(r^2 = x^2 + y^2\)): \((\sqrt{x^2 + y^2})^3 = xy\) This is the Cartesian equation equivalent to the given polar equation: \(x^2 + y^2)^{3/2} = xy\)

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Recommended explanations on Math Textbooks

Select your language

Chapter 30: Chapter 30

Convert the equation \(\mathrm{r}=\tan \theta+\cot \theta\) to an equation in cartesian coordinates.

Short Answer

Step by step solution

Unlock all solutions

Step 1: Rewrite the equation in terms of sine and cosine functions

Step 2: Find a common denominator and simplify the equation

Step 3: Substitute the polar coordinates with Cartesian coordinates

Step 4: Simplify the equation and solve for \(r\)

Step 5: Substitute back Cartesian coordinates for \(r\)

Access millions of textbook solutions in one place

Most popular questions from this chapter

More chapters from the book ‘Algebra and Trigonometry’

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

Chapter 25

Chapter 26

Chapter 27

Chapter 28

Chapter 29

Chapter 31

Chapter 32

Chapter 33

Chapter 34

Chapter 35

Chapter 36

Chapter 37

Chapter 38

Chapter 39

Chapter 40

Chapter 41

Chapter 42

Chapter 43

Chapter 44

Chapter 45

Chapter 46

Chapter 47

Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Decision Maths

Calculus

Pure Maths

Statistics