1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
zaharov [31]
3 years ago
12

The Cartesian coordinates of a point are given. (a) (−5, 5) (i) Find polar coordinates (r, θ) of the point, where r > 0 and 0

≤ θ < 2π. (r, θ) = (ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) = (b) (3, 3 3 ) (i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (r, θ) = (ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) =
Mathematics
1 answer:
Alex73 [517]3 years ago
6 0

Answer:

a)

(i) The coordinates of the point in polar form is (5√2 , 7π/4)

(ii) The coordinates of the point in polar form is (-5√2 , 3π/4)

b)

(i) The coordinates of the point in polar form is (6 , π/3)

(ii) The coordinates of the point in polar form is (-6 , 4π/3)

Step-by-step explanation:

* Lets study the meaning of polar form

- To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):

1. r = √( x2 + y2 )

2. θ = tan^-1 (y/x)

- In Cartesian coordinates there is exactly one set of coordinates for any

 given point

- In polar coordinates there is literally an infinite number of coordinates

 for a given point

- Example:

- The following four points are all coordinates for the same point.

# (5 , π/3) ⇒ 1st quadrant

# (5 , −5π/3) ⇒ 4th quadrant

# (−5 , 4π/3) ⇒ 3rd quadrant

# (−5 , −2π/3) ⇒ 2nd quadrant

- So we can find the points in polar form by using these rules:

 [r , θ + 2πn] , [−r , θ + (2n + 1) π] , where n is any integer

 (more than 1 turn)

* Lets solve the problem

(a)

∵ The point in the Cartesian plane is (-5 , 5)

∵ r = √x² + y²

∴ r = √[(5)² + (-5)²] = √[25 + 25] = √50 = ±5√2

∵ Ф = tan^-1 (y/x)

∴ Ф = tan^-1 (5/-5) = tan^-1 (-1)

- Tan is negative in the second and fourth quadrant

∵ 0 ≤ Ф < 2π

∴ Ф = 2π - tan^-1(1) ⇒ in fourth quadrant r > 0

∴ Ф = 2π - π/4 = 7π/4

OR

∴ Ф = π - tan^-1(1) ⇒ in second quadrant r < 0

∴ Ф = π - π/4 = 3π/4

(i) ∵ r > 0

∴ r = 5√2

∴ Ф = 7π/4 ⇒ 4th quadrant

∴ The coordinates of the point in polar form is (5√2 , 7π/4)

(ii) r < 0

∴ r = -5√2

∵ Ф = 3π/4 ⇒ 2nd quadrant

∴ The coordinates of the point in polar form is (-5√2 , 3π/4)

(b)

∵ The point in the Cartesian plane is (3 , 3√3)

∵ r = √x² + y²

∴ r = √[(3)² + (3√3)²] = √[9 + 27] = √36 = ±6

∵ Ф = tan^-1 (y/x)

∴ Ф = tan^-1 (3√3/3) = tan^-1 (√3)

- Tan is positive in the first and third quadrant

∵ 0 ≤ Ф < 2π

∴ Ф = tan^-1 (√3) ⇒ in first quadrant r > 0

∴ Ф = π/3

OR

∴ Ф = π + tan^-1 (√3) ⇒ in third quadrant r < 0

∴ Ф = π + π/3 = 4π/3

(i) ∵ r > 0

∴ r = 6

∴ Ф = π/3 ⇒ 1st quadrant

∴ The coordinates of the point in polar form is (6 , π/3)

(ii) r < 0

∴ r = -6

∵ Ф = 4π/3 ⇒ 3rd quadrant

∴ The coordinates of the point in polar form is (-6 , 4π/3)

You might be interested in
Pierre is running 26 and 2/10 miles in the marathon. He has run 3/4 of the way. How far has he run?
Harman [31]
Pierre ran 19 and 13/20 miles.

Given:
marathon length : 26 2/10 miles
pierre's length ran : 3/4 of the way.

Multiply the fractions. 
Convert the marathon length from a mixed fraction into a fraction.
26 2/10 = ((26*10)+2)/10 = (260+2)/10 = 262/10

262/10 * 3/4 = (262*3) / (10*4) = 786/40 = 19 26/40

19 26/40 simplified to 19 13/20 miles

26 ÷ 2 = 13
40 ÷ 2 = 20

3 0
3 years ago
A movie theater sends out a coupon for 70% off the price off a ticket. a) Write an equation for the situation, where y is the pr
sergey [27]
Equation 1
y=x-0.7

equation 2
x=y+0.7

4 0
3 years ago
Read 2 more answers
A store offers a 15% discount on a skirt. If Bernice pays $40.80 for the skirt, what was the list price of the skirt before the
nadya68 [22]
The answer to this question is: <span>x=48</span>
6 0
3 years ago
Read 2 more answers
324.56 × 54.82<br><br> SHOW YOUR WORK<br><br> ALL OF IT PLEASE AND THANK YOU!<br> PLEASE HURRY!
Mazyrski [523]

Answer:

17792.3792

Step-by-step explanation:

:))))))

8 0
2 years ago
2(a=4)=2a-8+4a solve for a
Naddik [55]

Answer:

a = 0.

Step-by-step explanation:

2(a - 4) = 2a - 8 + 4a

2a - 8 = 2a - 8 + 4a

2a - 2a - 4a = -8 + 8

-4a = 0

a = 0.

8 0
3 years ago
Other questions:
  • The figure shows a carpeted room. How many square feet of the room are carpeted?
    5·2 answers
  • What are the different types of triangles ?
    12·2 answers
  • What is -4 {2 - y} + 3y = 3 {y - 4}
    11·1 answer
  • We have 12 pumpkin pies that need to be shared evenly between 16 people
    9·1 answer
  • 1 point
    14·1 answer
  • The distance from C to D is 24 miles. The distance from B to C is 2/3 of the distance from C to D. The distance from A to B is 3
    7·1 answer
  • What are the solutions to x2 + x - 5≤0
    14·1 answer
  • Kim spent $33 on a magazine and
    14·2 answers
  • 2.25(a+1.3)=20.7 what is the value of a ? ​
    8·1 answer
  • What is the slope of the equation below? *
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!