## (-2,3π/2) = (2, π/2)

angle between them is 2π/3 - π/2 = π/6 ....... (think of it as 120° - 90° = 30°)

area = (1/2)(2)(sin π/6 )

= (1/2)(4)(1/2) = 1 square unit

(A) 1/2 (B) 1 (C) √2 (D) √3

mathhelper
answered

2 years ago

2 years ago

angle between them is 2π/3 - π/2 = π/6 ....... (think of it as 120° - 90° = 30°)

area = (1/2)(2)(sin π/6 )

= (1/2)(4)(1/2) = 1 square unit

Step-by-Step Bot
answered

1 year ago

1 year ago

Area = 1/2 * base * height

First, let's find the length of the base of the triangle. The base of the triangle is the distance between the two given points P and Q.

Using the distance formula in polar coordinates:

d = √((r2)^2 + (θ2 - θ1)^2)

where r1, r2 are the magnitudes (distances from the origin) of the two points, and θ1, θ2 are the angles of the two points.

For point P:

r1 = -2

θ1 = 3π/2

For point Q:

r2 = 2

θ2 = 2π/3

Calculating the distance (base) between P and Q:

d = √((2)^2 + (2π/3 - 3π/2)^2)

= √(4 + (4π/3 - 9π/6)^2)

= √(4 + (4π/3 - 3π/3)^2)

= √(4 + (π/3)^2)

= √(4 + π^2/9)

= √((36 + π^2)/9)

= (√(36 + π^2))/3

Now, let's find the height of the triangle. The height of the triangle is the distance between the pole and the line segment connecting P and Q.

Since the pole is at the origin (0,0), the distance from the origin to the line segment connecting P and Q is equal to the radius of either point.

Using the given points:

r1 = -2 (magnitude of point P)

r2 = 2 (magnitude of point Q)

Either r1 or r2 will give us the height of the triangle. Let's choose r1:

height = r1 = -2

Finally, let's calculate the triangular area using the formula:

Area = 1/2 * base * height

= 1/2 * (√(36 + π^2))/3 * (-2)

= -1/3 * (√(36 + π^2))

So, the triangular area formed by P, Q, and the pole is -1/3 * (√(36 + π^2)).

None of the answer choices match this result.

Please let me know if I can help you with anything else.

Explain Bot
answered

10 months ago

10 months ago

Let's convert the points P and Q from polar coordinates to Cartesian coordinates:

P(−2, 3π/2):

To convert polar coordinates to Cartesian coordinates, we use the formulas:

x = r * cos(θ)

y = r * sin(θ)

For P(−2, 3π/2):

r = -2

θ = 3π/2

x = -2 * cos(3π/2) = 0

y = -2 * sin(3π/2) = -2

So, the Cartesian coordinates of point P are (0, -2).

Q(2, 2π/3):

Using the same formulas for converting polar coordinates to Cartesian coordinates:

r = 2

θ = 2π/3

x = 2 * cos(2π/3) = -1

y = 2 * sin(2π/3) = √3

So, the Cartesian coordinates of point Q are (-1, √3).

Now, we have the Cartesian coordinates of points P and Q. Let's find the distance between these two points, which will be the base of the triangle formed by P, Q, and the pole.

Using the distance formula:

distance = √((x2 - x1)^2 + (y2 - y1)^2)

distance = √((0 - (-1))^2 + (-2 - √3)^2)

distance = √((1)^2 + (-2 - √3)^2)

distance = √(1 + 4 + √12 - 4√3)

distance = √(5 + √12 - 4√3)

Now, let's calculate the height of the triangle. The height is the y-coordinate of the pole, which is 0.

The area of a triangle is given by the formula: area = (base * height) / 2

Substituting the values:

area = ((√(5 + √12 - 4√3)) * 0) / 2

area = 0

Therefore, the area of the triangular region formed by P, Q, and the pole is 0.