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Question

Which one of the following does not represent the point (4, -75 deg) 
A. (-4, -615 deg) 
B. (4, -615 deg) 
C. (4, 285 deg) 
D. (4, 645 deg) 

Which of the following does not represent the point (-5, Pi/6) 
A. (-5, -11Pi/6) 
B. (-5, 13Pi/6) 
C. (5, 19Pi/6) 
D. (5, Pi/6) 

Which of the following does not represent the point (-2, -Pi/4) 
A. (-2, -9Pi/4) 
B. (-2, 15Pi/4) 
C. (2, 9Pi/4) 
D. (2, 11Pi/4) 

Please help to solve thanks.

Answer 1

(3).

The polar coordinate can be written as (r, θ) = (r, θ + 2nπ) or (r, θ) = [ - r, θ + (2n + 1)π ], where n is any integer.

The polar coordinate point (r, θ₁).

The radius r is same sign then angle θ₁ = θ + 2nπ and solve for n.

The value of n = (θ₁ - θ) / 2π where n must be integer.

The radius r opposite sign then angle θ₁ = θ + (2n + 1)π and solve for n.

The value of n = 1/2 { [ (θ₁ - θ) / π ] - 1 }, where n must be integer.

 

The given polar coordinate point (r, θ₁) = (- 2, - π/4).

 

The option A . (- 2, - 9π/4).

Here r = - 2 is same sign, so,  n = (θ₁ - θ) / 2π.

n = (- π/4 + 9π/4) / 2π

n = (8π/4) / 2π

n = 2π / 2π = 1 is an integer, so the point (- 2, - 9π/4) represents the point (- 2, - π/4).

 

The option B . (- 2, 15π/4).

Here r = - 2 is same sign, so,  n = (θ₁ - θ) / 2π.

n = (- π/4 - 15π/4) / 2π

n = (- 16π/4) / 2π

n = - 4π / 2π = - 2 is an integer, so the point (- 2, 15π/4) represents the point (- 2, - π/4).

 

The option C . (2, 9π/4).

Here r = 2 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / π ] - 1 }.

n = 1/2 { [ (- π/4 - 9π/4) / π ] - 1 }

= 1/2 { [ (- 10π/4) / π ] - 1 }

= 1/2 { - 5/2 - 1 }

= 1/2 {- 7/2 } = - 7/4 is not an integer, so the point (2, 9π/4) does not represent the point (- 2, - π/4).

 

The option D . (2, 11π/4).

Here r = 2 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / π ] - 1 }.

n = 1/2 { [ (- π/4 - 11π/4) / π ] - 1 }

= 1/2 { [ (- 12π/4) / π ] - 1 }

= 1/2 { [ - 3π / π ] - 1 }

= 1/2 { - 3 - 1 }

= 1/2 {- 4 } = - 2 is an integer, so the point (2, 9π/4) represents the point (- 2, - π/4).

 

The option C is correct answer.

Answer 2

(2).

The polar coordinate can be written as (r, θ) = (r, θ + 2nπ) or (r, θ) = [ - r, θ + (2n + 1)π ], where n is any integer.

The polar coordinate point (r, θ₁).

The radius r is same sign then angle θ₁ = θ + 2nπ and solve for n.

The value of n = (θ₁ - θ) / 2π where n must be integer.

The radius r opposite sign then angle θ₁ = θ + (2n + 1)π and solve for n.

The value of n = 1/2 { [ (θ₁ - θ) / π ] - 1 }, where n must be integer.

 

The given polar coordinate point (r, θ₁) = (- 5, π/6).

 

The option A . (- 5, - 11π/6).

Here r = - 5 is same sign, so,  n = (θ₁ - θ) / 2π.

n = (π/6 + 11π/6) / 2π

n = (12π/6) / 2π

n = 2π / 2π = 1 is an integer, so the point (- 5, - 11π/6) represents the point (- 5, π/6).

 

The option B . (- 5, 13π/6).

Here r = - 5 is same sign, so,  n = (θ₁ - θ) / 2π.

n = (π/6 - 13π/6) / 2π

n = (- 12π/6) / 2π

n = - 2π / 2π = - 1 is an integer, so the point (- 5, 13π/6) represents the point (- 5, π/6).

 

The option C . (5, 19π/6).

Here r = 5 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / π ] - 1 }.

n = 1/2 { [ (π/6 - 19π/6) / π ] - 1 }

= 1/2 { [ (- 18π/6) / π ] - 1 }

= 1/2 { [- 3π / π ] - 1 }

= 1/2 {- 3 - 1 } = - 2 is an integer, so the point (5, 19π/6) represents the point (- 5, π/6).

 

The option D . (5, π/6).

Here r = 5 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / π ] - 1 }.

n = 1/2 { [ (π/6 - π/6) / π ] - 1 }

= 1/2 { [0] - 1 } = - 1/2 is not an integer, so the point (5, π/6) does not represent the point (- 5, π/6).

 

The option D is correct answer.

Answer 3

(1).

The polar coordinate can be written as (r, θ) = (r, θ + 2nπ) or (r, θ) = [ - r, θ + (2n + 1)π ], where n is any integer.

The polar coordinate point (r, θ) = (4, - 75^o).

The option A . (- 4, - 615^o).

The radius r = - 4 (opposite sign) and the angle - 615 can be written as θ + (2n + 1)π = - 615^o + [2(1) + 1] 180^o = - 75^o, since n = 1 is an integer.

The option B . (4, - 615^o).

The radius r = 4 (same sign) and the angle - 615 cannot be written as θ + 2nπ = - 615^o + 2(3/2) 180^o = - 75^o, since n = 3/2 is not an integer.

The option C . (4, 285^o).

The radius r = 4 (same sign) and the angle 285 cannot be written as θ + 2nπ = 285^o + 2(- 1) 180^o = - 75^o, since n =  - 1 is an integer.

The option D . (4, 645^o).

The radius r = 4 (same sign) and the angle 285 cannot be written as θ + 2nπ = 645^o + 2(- 2) 180^o = - 75^o, since n =  - 2 is an integer.

The option B is correct answer.