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Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
ISBN: 9781305071759

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BuyFindarrow_forward

Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
ISBN: 9781305071759
Textbook Problem

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

(a)

To determine

To describe: The polar coordinate system.

Explanation

The coordinate system (r,θ) signifies the polar coordinates with r as radius and θ as the angle between the polar axis and the line segment joining the point.

In polar coordinate system r shows the distance of the point and θ shows the direction of the polar coordinate (r,θ) .

In polar coordinate system take θ positive in counter clockwise direction else take θ negative in clockwise direction.

In polar coordinate system negative r signifies that the polar coordinate (r,θ) lies |r| units from the pole in opposite direction of angle θ .

The below figure shows the polar coordinates (r,θ) .

images

Figure (1)

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

(b)

To determine

To sketch: The graph of the polar coordinates.

Explanation

The below graph shows the polar coordinates (2,π3) and (1,3π4) .

images

Figure (2)

In the above graph, point P(2,π3) and Q(1,3π4) show the required points in polar coordinate system.

(c)

To determine

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

Explanation

Use the equations x=rcosθ and y=rsinθ to convert the polar coordinate system of an equation to its corresponding rectangular coordinate system.

Use the equations r2=x2+y2 and tanθ=yx(x0) to convert rectangular coordinate system of a point to its corresponding polar coordinate system.

(d)

To determine

To find: The rectangular coordinate of the point.

Answer

The rectangular coordinate of the point (2,π3) is (1,3) .

Explanation

Given:

The value of polar coordinate is (2,π3) .

Calculation:

Use the equations x=rcosθ and y=rsinθ to convert the polar coordinate system of a equation to its corresponding rectangular coordinate system.

The formula to calculate the x coordinate is,

x=rcosθ .

Substitute 2 for r and π3 for θ in the above formula.

x=2cosπ3=2×12=1

The value of the x coordinate is 1.

The formula to calculate the y coordinate is,

y=rsinθ .

Substitute 2 for r and π3 for θ in the above formula.

y=2sinπ3=2×32=3

The value of the y coordinate is 3 .

Thus, the rectangular coordinate of the point (2,π3) is (1,3) .

(e)

To determine

To find: The polar coordinate of the point.

Answer

The polar coordinate of the point (2,2) is (22,π4) .

Explanation

Given:

The value of rectangular coordinate is (2,2) .

Calculation:

Use the equations r2=x2+y2 and tanθ=yx(x0) to convert rectangular coordinate system of a point to its corresponding polar coordinate system.

The formula to calculate the r is,

r2=x2+y2

Substitute 2  for x and 2 for y in the above formula.

r2=(2)2+(2)2=4+4=8=22

The value of r is 22 .

The formula to calculate the value of θ is,

tanθ=yx(x0)

Substitute 2 for x and 2 for y in the above formula,

tanθ=22=1tan1(tanθ)=tan1(1)(Taketan1θboth sides)θ=π4(tan1(tanθ)=θ)

The value of θ is π4 .

Thus, the rectangular coordinate of the point (2,2) is (22,π4) .

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