convert each point,P(x,y)to the ordered pair P(r cos theta ,r sin theta).round all valuas of r to one dicimal place and all values of theta to the nearste degree.
(a) (4,6)
(b) (-3.7)
(c) (10,-23)
I will do a), you do the rest
for (4,6), r^2= 4^2+ 6^2 = 52
r= √52
tanØ= 6/4
Ø = appr 56.3°
so (4,6) ---> (√52cos56.3 , √52sin56.3)
= appr (7.2cos56° , 7.2sin56°)
AR U SURE ?
Did you want the ordered pair (4,6) expressed in polar form?
I was wondering why you asked for the form
(rcosØ, rsinØ) since that is simply the equivalent of (x,y)
e.g. my answer of (7.2cos56 , 7.2sin56) = (4.026 , 5.969) or (4,6)
if you want it in the polar form (r, Ø)
then (4,6) ---> (7.2 , 56°)
however, that is not what you had asked for.
To convert each point, P(x,y), to the ordered pair P(r cosθ, r sinθ), we use the polar coordinate system. The polar coordinates consist of the radial distance (r) from the origin to the point and the angle (θ) measured counterclockwise from the positive x-axis to the line segment connecting the origin and the point.
To convert a point from rectangular coordinates (x,y) to polar coordinates (r,θ), you can follow these steps:
Step 1: Calculate the value of r:
r = √(x^2 + y^2)
where x is the x-coordinate and y is the y-coordinate of the given point.
Step 2: Calculate the value of θ:
θ = arctan(y/x)
where x is the x-coordinate and y is the y-coordinate of the given point.
Step 3: Round the values of r to one decimal place and the values of θ to the nearest degree.
Now, let's use these steps to convert each of the given points to polar coordinates:
(a) (4,6)
Step 1: Calculate r:
r = √(4^2 + 6^2) = √(16 + 36) = √52 ≈ 7.2
Step 2: Calculate θ:
θ = arctan(6/4) = arctan(1.5) ≈ 56° (rounded to the nearest degree)
So, P(4,6) in polar coordinates is approximately P(7.2 cos 56°, 7.2 sin 56°).
(b) (-3,7.5)
Step 1: Calculate r:
r = √((-3)^2 + 7.5^2) = √(9 + 56.25) = √65.25 ≈ 8.1
Step 2: Calculate θ:
θ = arctan(7.5/-3) = arctan(-2.5) ≈ -68° (rounded to the nearest degree)
So, P(-3,7.5) in polar coordinates is approximately P(8.1 cos -68°, 8.1 sin -68°).
(c) (10,-23)
Step 1: Calculate r:
r = √(10^2 + (-23)^2) = √(100 + 529) = √629 ≈ 25.1
Step 2: Calculate θ:
θ = arctan((-23)/10) = arctan(-2.3) ≈ -68° (rounded to the nearest degree)
So, P(10,-23) in polar coordinates is approximately P(25.1 cos -68°, 25.1 sin -68°).
Remember to round the values of r and θ as mentioned in the question.