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
Crank
3 years ago
9

The Cartesian coordinates of a point are given. (a) (4, −4)(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0

≤ θ < 2π.(ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π.(b) (−1, sqrt(3))(i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π.(ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π.
Mathematics
1 answer:
Whitepunk [10]3 years ago
7 0

Answer:

(x,y) = (r,\theta)\\

i)(4,-4) = (\pm 4\sqrt{2},\dfrac{7 \pi}{4})\\

ii) (-1, \sqrt{3}) = (\pm 2,\dfrac{2 \pi}{4})\\

Thinking Process:

This can either be solved using complex variables or simply trigonometry (they are all the same in a way)

(a,b) = a + ib = r(\cos{(\theta)} + i\sin{(\theta)}) = r e^{(i \theta)} = r\angle{\theta}

But we'll simply go with trigonometry and good old Pythagorean theorem.

Solution:

For Cartesian coordinates are: (x,y)

To covert them into r \angle{\theta}

We can use:

  • The Pythagoras theorem to find r

r = \pm \sqrt{x^2 + y^2}

  • Arctangent to find \theta

\theta = \tan^{-1}{\left(\dfrac{y}{x}\right)}

These are related since:

x = r\cos{(\theta)}

y = r\sin{(\theta)}

So, let's start solving:

Part a) (4,-4)

For r:

r = \pm \sqrt{x^2 + y^2}

r = \pm \sqrt{4^2 + (-4)^2}

r = \pm \sqrt{32}

r = \pm 4 \sqrt{2}

i) for r > 0 we'll go with r = + 4\sqrt{2}

ii) for r < 0 we'll go with r = - 4\sqrt{2}

For theta:

\theta = \tan^{-1}{\left(\dfrac{y}{x}\right)}

\theta = \tan^{-1}{\left(\dfrac{-4}{4}\right)}

\theta = \tan^{-1}{(-1)}

\theta = -45\texdegree \,\, or -\dfrac{\pi}{4} radians

One thing to keep in mind is that the point (4,-4) lies in the 4th quadrant of the xy-plane, and the range given for \theta is 0 \leq \theta \leq 2 \pi . So we have to add 2 \pi to our answer (we have gone around the circle and stopped at the same place as -\frac{\pi}{4} radians)

\theta=-\dfrac{\pi}{4} + 2 \pi radians

\theta=\dfrac{7\pi}{4} radians

The Answer for part(a) is (\pm 4\sqrt{2}, \dfrac{7 \pi}{4})

Part b) (-1, sqrt(3))

For r:

r = \pm \sqrt{x^2 + y^2}

r = \pm \sqrt{(-1)^2 + (\sqrt{3})^2}

r = \pm \sqrt{(-1)^2 + (\sqrt{3})^2}

i) r = +2 for r > 0

ii) r = -2 for r < 0

For theta:

\theta = \tan^{-1}{\left(\dfrac{y}{x}\right)}

\theta = \tan^{-1}{\left(\dfrac{\sqrt{3}}{-1}\right)}

\theta = -\dfrac{\pi}{3} this not in range of theta: 0 \leq \theta \leq 2 \pi.

and the point (-1, \sqrt{3}) lies in the 2nd quadrant, so we can add \pi radians to our answer

\theta = -\dfrac{\pi}{3} + \pi in range: 0 \leq \theta \leq 2 \pi

\theta = \dfrac{2 \pi}{3}

The Answer for part(b) is (\pm 2, \dfrac{2 \pi}{3})

You might be interested in
6th grade math plz help me
anastassius [24]
1) 8 41/48
2) 9 16/27
3) 41 2/5
4) 13 55/63
hope this helps x
3 0
3 years ago
X-23 / x^2 -x- 20 - 2/5-x
alexira [117]

By simplifying \[x - \frac{{23}}{{{x^2}}} - x - 20 - \frac{2}{5} - x\] . This will result in a simplified version of  \[ - \frac{{5{x^3} + 102{x^2} + 115}}{{5{x^2}}}\].

The Simplifying Algorithm is a wonderful way to simplify complex mathematics problems. It can be used to solve equations, convert fractions to decimals, and perform many other math operations. In this problem, the Simplifying Algorithm will help you reduce \[x - \frac{{23}}{{{x^2}}} - x - 20 - \frac{2}{5} - x\]

Since two opposites add up to 0, remove them from the expression.

\[ - \frac{{23}}{{{x^2}}} - \frac{{102}}{5} - x\]

Write all numerators above the least common denominator 5x2

\[ - \frac{{115 + 102{x^2} + 5{x^3}}}{{5{x^2}}}\]

Use the commutative property to reorder the terms so that constants on the left

\[\frac{{ - 5{x^3} - 115 - 102{x^2}}}{{5{x^2}}}\]

Rearrange the terms

\[\frac{{ - 5{x^3} - 102{x^2} - 115}}{{5{x^2}}}\]

By reording the terms

\[ - \frac{{5{x^3} + 102{x^2} + 115}}{{5{x^2}}}\]

Hence, by simplifying this equation, divide both numerator and denominator. This will result in a simplified version of  \[ - \frac{{5{x^3} + 102{x^2} + 115}}{{5{x^2}}}\].

To learn more about simplifyication visit:

brainly.com/question/1542396

#SPJ1

7 0
1 year ago
Shipping rates for Company A and Company B are shown in the tables below. Which company has shipping rates that you can represen
meriva

Company B; the ratios of cost to weight are equivalent.

Step-by-step explanation:

Step 1:

In the equation, y=kx k is the constant of proportionality.

If the values are in accordance with y=kx, the values of k will be constant for all the values.

So we determine the values of k for both the companies and see which has a constant k.

If y=kx, k = \frac{y}{x}. In these tables, y is the total cost and x is the weight in lbs.

Step 2:

For company A,

when y=10.55, x=1, k = \frac{10.55}{1} = 10.55,

when y=10.85, x=2, k = \frac{10.85}{2} = 5.425,

when y=11.15, x=3, k = \frac{11.15}{3} = 3.71666.

For company B,

when y=2.75, x=1, k = \frac2.75}{1} = 2.75,

when y=5.50, x=2, k = \frac{5.50}{2} = 2.75,

when y=8.25, x=3, k = \frac{8.25}{3} = 2.75.

So company B has a constant value of k=2.75.

3 0
3 years ago
Help me...............​
True [87]

Answer:

Brainliestgive o020201000

3 0
3 years ago
Factor x2 + 29x - 30.
marysya [2.9K]
Ok the w to your question is to cross multiply <span>216=9m m=24

</span>
8 0
3 years ago
Read 2 more answers
Other questions:
  • Madison drew this shape how many angles less than a right angle does it have?
    7·1 answer
  • What is 3.868x10^9 in standard form
    9·1 answer
  • a sequence of numbers begins at 400. If the pattern increases by consecutive multiples of 5 starting with 5, then find the next
    12·1 answer
  • over the course of four days the price of stock at ACM E corporation fall $28. what was the average change in stock price per da
    15·1 answer
  • What is the difference between the volume of a cone and the volume of the cylinder
    6·1 answer
  • In a lab experiment, 130 bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to dou
    12·1 answer
  • Is 14 a rational number.?
    9·1 answer
  • Find AB. Round to the nearest tenth if necessary.<br>​
    10·1 answer
  • Find the area of AAOB<br> 9√3 un²<br> 3√3 un²<br> 4.5√3 un
    14·1 answer
  • Which is the graph of 5 log (x + 3)?
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!