I don't understand how to do these problems:
Find two pairs of polar coordinates for each point with the given rectangular coordinates if 0 ≤ θ ≤ 2π. Round to the nearest hundredth, if necessary.
17. (2, -3)
Answers: (3.61, 5.30) and (-3.61, 2.16)
19. (a, 3a), a > 0
Answers: (3.16a, 1.25) and (-3.16a, 4.39)
Thank you!!
make a sketch and place (2,-3) into the fourth quadrant. Complete the right-angled triangle
the hypotenuse will be √13 , x - 2, y = -3
let the angle be Ø
tanØ = -3/2
Ø = 2π - .98279.. = 5.3
so one point is (√13, 5.3) which is what you have.
your other point is also correct.
for the point (a,3a) , a > 0 , the point is in quad I
tanØ = 3a/a = 3
Ø = 1.25 , you are right
r = √(a^2 + 9a^) = a√10 = appr 3.16a
again, you are right, good job
To find the polar coordinates for a point given in rectangular coordinates, we can use the following formulas:
r = √(x^2 + y^2)
θ = arctan(y / x)
For each point, let's first calculate the value of r:
17. (2, -3):
r = √(2^2 + (-3)^2)
= √(4 + 9)
= √13
19. (a, 3a):
r = √(a^2 + (3a)^2)
= √(a^2 + 9a^2)
= √(10a^2)
= √10 * a
Next, we'll calculate the value of θ. To do this, we need to consider the signs of x and y:
17. (2, -3):
Since x is positive and y is negative, we can use the formula θ = arctan(y / x) directly. In this case,
θ = arctan((-3) / 2)
= -56.31° (rounded to two decimal places)
19. (a, 3a):
Since x is positive and y is positive, we can use the formula θ = arctan(y / x) directly. In this case,
θ = arctan((3a) / a)
= arctan(3)
= 71.57° (rounded to two decimal places)
Now, we have the values of r and θ for each point. Remember that θ should be in the range of 0 ≤ θ ≤ 2π. To find the second pair of polar coordinates, we can add 180° to θ:
17. (2, -3):
Second pair: (-56.31° + 180°)
= 123.69° (rounded to two decimal places)
19. (a, 3a):
Second pair: (71.57° + 180°)
= 251.57° (rounded to two decimal places)
Finally, we can convert the polar coordinates back to rectangular coordinates using the formulas:
x = r * cos(θ)
y = r * sin(θ)
17. (2, -3):
x = √13 * cos(-56.31°)
= 3.61 (rounded to two decimal places)
y = √13 * sin(-56.31°)
= -5.30 (rounded to two decimal places)
Second pair: (-3.61, 5.30)
19. (a, 3a):
x = √10 * a * cos(71.57°)
= 3.16a (rounded to two decimal places)
y = √10 * a * sin(71.57°)
= 1.25a (rounded to two decimal places)
Second pair: (-3.16a, 1.25a)
Therefore, the two pairs of polar coordinates for each point are:
17. (2, -3):
(3.61, 5.30) and (-3.61, 2.16)
19. (a, 3a):
(3.16a, 1.25a) and (-3.16a, 4.39a)
Remember to round the answers to the nearest hundredth, if necessary.