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CalculusPrecalculus: Mathematics for Calculus (Standalone Book)(a) Explain the polar coordinate system. (b) Graph the points with polar coordinates (2, π /3) and (−1, 3 π /4). (c) State the equations that relate the rectangular coordinates of a point to its polar coordinates. (d) Find rectangular coordinates for (2, π /3). (e) Find polar coordinates for P (−2, 2).BuyFind*arrow_forward*

7th Edition

James Stewart + 2 others

Publisher: Brooks Cole

ISBN: 9781305071759

Chapter 8, Problem 1RCC

Textbook Problem

**(a)**Explain the polar coordinate system.**(b)**Graph the points with polar coordinates (2,*π*/3) and (−1, 3*π*/4).**(c)**State the equations that relate the rectangular coordinates of a point to its polar coordinates.**(d)**Find rectangular coordinates for (2,*π*/3).**(e)**Find polar coordinates for*P*(−2, 2).

Expert Solution

**(a)**

To determine

**To describe:** The polar coordinate system.

The coordinate system
*r* as radius and

In polar coordinate system *r* shows the distance of the point and

In polar coordinate system take

In polar coordinate system negative *r* signifies that the polar coordinate

The below figure shows the polar coordinates

**Figure (1)**

In the above figure, the point *P* is *r* unit away adjoining with angle

Expert Solution

**(b)**

To determine

**To sketch:** The graph of the polar coordinates.

The below graph shows the polar coordinates

**Figure (2)**

In the above graph, point

Expert Solution

**(c)**

To determine

**To describe:** The equations that relate the rectangular coordinates and polar coordinates of a point with each other.

Use the equations

Use the equations

Expert Solution

**(d)**

To determine

**To find:** The rectangular coordinate of the point.

The rectangular coordinate of the point

**Given:**

The value of polar coordinate is

**Calculation:**

Use the equations

The formula to calculate the *x* coordinate is,

Substitute 2 for *r* and

The value of the *x* coordinate is 1.

The formula to calculate the *y* coordinate is,

Substitute 2 for *r* and

The value of the *y* coordinate is

Thus, the rectangular coordinate of the point

Expert Solution

**(e)**

To determine

**To find:** The polar coordinate of the point.

The polar coordinate of the point

**Given:**

The value of rectangular coordinate is

**Calculation:**

Use the equations

The formula to calculate the *r* is,

Substitute
*x* and

The value of *r* is

The formula to calculate the value of

Substitute
*x* and

The value of

Thus, the rectangular coordinate of the point

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Ch. 8.1 - We can describe the location of a point in the...Ch. 8.1 - Let P be a point in the plane. (a) If P has polar...Ch. 8.1 - Yes or No? If No, give a reason. 3. Do the polar...Ch. 8.1 - Yes or No? If No, give a reason. 4. Do the...Ch. 8.1 - Plotting Points in Polar Coordinates Plot the...Ch. 8.1 - Plotting Points in Polar Coordinates Plot the...Ch. 8.1 - Plotting Points in Polar Coordinates Plot the...Ch. 8.1 - Plotting Points in Polar Coordinates Plot the...Ch. 8.1 - Plotting Points in Polar Coordinates Plot the...Ch. 8.1 - Plotting Points in Polar Coordinates Plot the...

Ch. 8.1 - Different Polar Coordinates for the Same Point...Ch. 8.1 - Different Polar Coordinates for the Same Point...Ch. 8.1 - Different Polar Coordinates for the Same Point...Ch. 8.1 - Different Polar Coordinates for the Same Point...Ch. 8.1 - Different Polar Coordinates for the Same Point...Ch. 8.1 - Different Polar Coordinates for the Same Point...Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Rectangular Coordinates to Polar Coordinates A...Ch. 8.1 - Rectangular Coordinates to Polar Coordinates A...Ch. 8.1 - Polar Coordinates to Rectangular Coordinates A...Ch. 8.1 - 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Rectangular Equations to Polar Equations Convert...Ch. 8.1 - Rectangular Equations to Polar Equations Convert...Ch. 8.1 - Rectangular Equations to Polar Equations Convert...Ch. 8.1 - Rectangular Equations to Polar Equations Convert...Ch. 8.1 - Rectangular Equations to Polar Equations Convert...Ch. 8.1 - Rectangular Equations to Polar Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - Polar Equations to Rectangular Equations Convert...Ch. 8.1 - DISCUSS PROVE: The Distance Formula in Polar...Ch. 8.1 - DISCUSS: Different Coordinate Systems As was noted...Ch. 8.2 - To plot points in polar coordinates, we use a grid...Ch. 8.2 - (a) To graph a polar equation r = f(), we plot all...Ch. 8.2 - Graphs of Polar Equations Match the polar equation...Ch. 8.2 - Graphs of Polar Equations Match the polar equation...Ch. 8.2 - Graphs of Polar Equations Match the polar equation...Ch. 8.2 - Graphs of Polar Equations Match the polar equation...Ch. 8.2 - 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(a) The modulus of z is r =...Ch. 8.3 - (a) The complex number z = 1 + i in polar form is...Ch. 8.3 - How many different nth roots does a nonzero...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - Graphing Complex Numbers Sketch the complex...Ch. 8.3 - Graphing Complex Numbers Sketch the complex...Ch. 8.3 - Graphing a Complex Number and Its Complex...Ch. 8.3 - Graphing a Complex Number and Its Complex...Ch. 8.3 - 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Products and Quotients of Complex Numbers Find the...Ch. 8.3 - Products and Quotients of Complex Numbers Find the...Ch. 8.3 - Products and Quotients of Complex Numbers Find the...Ch. 8.3 - Products and Quotients of Complex Numbers Find the...Ch. 8.3 - Products and Quotients of Complex Numbers Write z1...Ch. 8.3 - Products and Quotients of Complex Numbers Write z1...Ch. 8.3 - Products and Quotients of Complex Numbers Write z1...Ch. 8.3 - Products and Quotients of Complex Numbers Write z1...Ch. 8.3 - Products and Quotients of Complex Numbers Write z1...Ch. 8.3 - Products and Quotients of Complex Numbers Write z1...Ch. 8.3 - Products and Quotients of Complex Numbers Write z1...Ch. 8.3 - Products and Quotients of Complex Numbers Write z1...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Powers Using De Moivres Theorem Find the indicated...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Solving Equations Using nth Roots Solve the...Ch. 8.3 - Solving Equations Using nth Roots Solve the...Ch. 8.3 - Solving Equations Using nth Roots Solve the...Ch. 8.3 - Solving Equations Using nth Roots Solve the...Ch. 8.3 - Solving Equations Using nth Roots Solve the...Ch. 8.3 - Solving Equations Using nth Roots Solve the...Ch. 8.3 - Complex Coefficients and the Quadratic Formula The...Ch. 8.3 - Complex Coefficients and the Quadratic Formula The...Ch. 8.3 - Complex Coefficients and the Quadratic Formula The...Ch. 8.3 - Complex Coefficients and the Quadratic Formula The...Ch. 8.3 - Finding nth roots of a Complex Number Let w =...Ch. 8.3 - Finding nth roots of a Complex Number Let w =...Ch. 8.3 - DISCUSS: Sums of Roots of Unity Find the exact...Ch. 8.3 - DISCUSS: Products of Roots of Unity Find the...Ch. 8.3 - PROOF: Division in Polar Form If the two complex...Ch. 8.4 - (a) The parametric equations x = f(t) and y = g(t)...Ch. 8.4 - (a) True or False? The same curve can be described...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Circular Motion The position of an object in...Ch. 8.4 - Circular Motion The position of an object in...Ch. 8.4 - Circular Motion The position of an object in...Ch. 8.4 - Circular Motion The position of an object in...Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Path of a Projectile If a projectile is fired with...Ch. 8.4 - Path of a Projectile Referring to Exercise 37,...Ch. 8.4 - Graphs of Parametric Equations Use a graphing...Ch. 8.4 - Graphs of Parametric Equations Use a graphing...Ch. 8.4 - Graphs of Parametric Equations Use a graphing...Ch. 8.4 - Graphs of Parametric Equations Use a graphing...Ch. 8.4 - Graphs of Parametric Equations Use a graphing...Ch. 8.4 - Graphs of Parametric Equations Use a graphing...Ch. 8.4 - Parametric Form of a Polar Equation A polar...Ch. 8.4 - Parametric Form of a Polar Equation A polar...Ch. 8.4 - Parametric Form of a Polar Equation A polar...Ch. 8.4 - Parametric Form of a Polar Equation A polar...Ch. 8.4 - Graphs of Parametric Equations Match the...Ch. 8.4 - Graphs of Parametric Equations Match the...Ch. 8.4 - Graphs of Parametric Equations Match the...Ch. 8.4 - Graphs of Parametric Equations Match the...Ch. 8.4 - Finding Parametric Equations for a Curve Two...Ch. 8.4 - Finding Parametric Equations for a Curve Two...Ch. 8.4 - Curtate Cycloid (a) In Example 6, suppose the...Ch. 8.4 - Prolate Cycloid (a) In Exercise 55 if the point P...Ch. 8.4 - Parametric Equations of a Hyperbola Eliminate the...Ch. 8.4 - Parametric Equations of a Hyperbola Show that the...Ch. 8.4 - Graphs of Parametric Equations Sketch the curve...Ch. 8.4 - Graphs of Parametric Equations Sketch the curve...Ch. 8.4 - Graphs of Parametric Equations Sketch the curve...Ch. 8.4 - Graphs of Parametric Equations Sketch the curve...Ch. 8.4 - Hypocycloid A circle C of radius b rolls on the...Ch. 8.4 - Epicycloid If the circle C of Exercise 63 rolls on...Ch. 8.4 - Longbow Curve In the following figure, the circle...Ch. 8.4 - The Witch of Agnesi A curve, called a witch of...Ch. 8.4 - Eliminating the Parameter Eliminate the parameter ...Ch. 8.4 - The Rotary Engine The Mazda RX-8 uses an...Ch. 8.4 - Spiral Path of a Dog A dog is lied to a...Ch. 8.4 - DISCOVER WRITE: More Information in Parametric...Ch. 8.4 - DISCUSS: Different Ways of Tracing Out a Curve The...Ch. 8 - (a) Explain the polar coordinate system. (b) Graph...Ch. 8 - (a) What is a polar equation? (b) Convert the...Ch. 8 - (a) How do we graph a polar equation? (b) Sketch a...Ch. 8 - (a) What is the complex plane? How do we graph a...Ch. 8 - (a) How do we express the complex number z in...Ch. 8 - Let z1=2(cos3+isin3)andz2=5(cos4+isin4) (a) Find...Ch. 8 - (a) State De Moivres Theorem. (b) Use De Moivres...Ch. 8 - (a) State the formula for the nth roots of a...Ch. 8 - (a) What are parametric equations? (b) Sketch a...Ch. 8 - Polar Coordinates to Rectangular Coordinates A...Ch. 8 - Polar Coordinates to Rectangular Coordinates A...Ch. 8 - Polar Coordinates to Rectangular Coordinates A...Ch. 8 - Polar Coordinates to Rectangular Coordinates A...Ch. 8 - Polar Coordinates to Rectangular Coordinates A...Ch. 8 - Polar Coordinates to Rectangular Coordinates A...Ch. 8 - Rectangular Coordinates to Polar Coordinates A...Ch. 8 - Rectangular Coordinates to Polar Coordinates A...Ch. 8 - Rectangular Coordinates to Polar Coordinates A...Ch. 8 - Rectangular Coordinates to Polar Coordinates A...Ch. 8 - Rectangular Coordinates to Polar Coordinates A...Ch. 8 - Rectangular Coordinates to Polar Coordinates A...Ch. 8 - Rectangular Equations to Polar Equations (a)...Ch. 8 - Rectangular Equations to Polar Equations (a)...Ch. 8 - Rectangular Equations to Polar Equations (a)...Ch. 8 - Rectangular Equations to Polar Equations (a)...Ch. 8 - Polar Equations to Rectangular Equations (a)...Ch. 8 - Polar Equations to Rectangular Equations (a)...Ch. 8 - Polar Equations to Rectangular Equations (a)...Ch. 8 - Polar Equations to Rectangular Equations (a)...Ch. 8 - Polar Equations to Rectangular Equations (a)...Ch. 8 - Polar Equations to Rectangular Equations (a)...Ch. 8 - Polar Equations to Rectangular Equations (a)...Ch. 8 - Polar Equations to Rectangular Equations (a)...Ch. 8 - Graphing Polar Equations Use a graphing device to...Ch. 8 - Graphing Polar Equations Use a graphing device to...Ch. 8 - Graphing Polar Equations Use a graphing device to...Ch. 8 - Graphing Polar Equations Use a graphing device to...Ch. 8 - Complex Numbers A complex number is given. (a)...Ch. 8 - Complex Numbers A complex number is given. (a)...Ch. 8 - Complex Numbers A complex number is given. (a)...Ch. 8 - Complex Numbers A complex number is given. (a)...Ch. 8 - Complex Numbers A complex number is given. (a)...Ch. 8 - Complex Numbers A complex number is given. (a)...Ch. 8 - Powers Using De Moivres Theorem Use De Moivres...Ch. 8 - Powers Using De Moivres Theorem Use De Moivres...Ch. 8 - Powers Using De Moivres Theorem Use De Moivres...Ch. 8 - Powers Using De Moivres Theorem Use De Moivres...Ch. 8 - Roots of Complex Numbers Find the indicated roots....Ch. 8 - Roots of Complex Numbers Find the indicated roots....Ch. 8 - Roots of Complex Numbers Find the indicated roots....Ch. 8 - Roots of Complex Numbers Find the indicated roots....Ch. 8 - Parametric Curves A pair of parametric equations...Ch. 8 - Parametric Curves A pair of parametric equations...Ch. 8 - Parametric Curves A pair of parametric equations...Ch. 8 - Parametric Curves A pair of parametric equations...Ch. 8 - Graphs of Parametric Equations Use a graphing...Ch. 8 - Graphs of Parametric Equations Use a graphing...Ch. 8 - Finding Parametric Equations for a Curve In the...Ch. 8 - (a) Convert the point whose polar coordinates are...Ch. 8 - (a) Graph the polar equation r = 8 cos . What type...Ch. 8 - Graph the polar equation r = 3 + 6 sin . What type...Ch. 8 - Let z=1+3i. (a) Graph z in the complex plane. (b)...Ch. 8 - Let z1=4(cos712+isin712) and z2=2(cos512+isin512)....Ch. 8 - Find the cube roots of 27i, and sketch these roots...Ch. 8 - (a) Sketch the curve represented by the parametric...Ch. 8 - Find parametric equations for the line of slope 2...Ch. 8 - The position of an object in circular motion is...Ch. 8 - Trajectories Are Parabolas From the graphs in...Ch. 8 - Path of a Baseball Suppose a baseball is thrown at...Ch. 8 - Path of a Rocket Suppose that a rocket is fired at...Ch. 8 - Firing a Missile The initial speed of a missile is...Ch. 8 - Maximum Height Show that the maximum height...Ch. 8 - Shooting into the Wind Suppose that a projectile...Ch. 8 - Shooting into the Wind Using the parametric...

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The Taylor polynomial of degree 3 for f(x) = x(ln x − 1) about 1 is T3(x) =
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In Exercises 12 to 14, write the first statement of the indirect proof of the given statement. If ACBC in ABC, ...

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Finding the Mass of a Lamina In Exercises 3-6, find the mass of the lamina described by the inequalities, given...

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As part of a study on transportation safety, the U.S. Department of Transportation collected data on the number...

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The article The Value and the Limitations of High-Speed Turbo-Exhausters for the Removal of Tar-Fog from Carbur...

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Using the Table of Block Thicknesses for a Metric Gage Block Set, determine a combination of gage blocks each o...

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Practice Find the value of b, if any, that would cause the graph of fx=bx to look like the graph indicated.

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Refer to the data in exercise 31. Consider a model in which only Industry is used to predict Delay. At a .01 le...

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Use the theorem on polynomial orders to prove each of the statements in 26—31. 29. n(n1)(2n+1)6 is (n3)

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In Exercises 2128, express the given biconditionals as the conjunction of two conditionals. A triangle having a...

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Applications Exercises 5 through 13 illustrate applications of optimization using parabolas. Size of High Schoo...

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Use (he following information to answer the next four exercises. A box Is filled with several party favors. It ...

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For the following exercises, consider an athlete running a 40-m dash. The position of the athlete is given by d...

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