Convert the polar coordinates (-83,23°) into rectangular coordinates. Round the trigonometric values and the rectangular coordinates to the nearest hundredth.
just plug in your transformations
x = r cosθ
y = r sinθ
-83[23o] = -83*Cos23 + (-83*sin23)i = -76.40 - 32.43i.
Thanks Oobleck and Henry2 I got it now!
actually, you don't need the "i" stuff unless you want to represent the point as a complex number.
the polar point (-83,23°) is the x-y point (-76.40, - 32.43)
To convert polar coordinates to rectangular coordinates, we can use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
where:
r is the radius or distance from the origin to the point,
θ is the angle (in degrees) between the positive x-axis and the ray from the origin to the point.
In this case, the polar coordinates are given as (-83, 23°).
Step 1: Convert the angle from degrees to radians.
To use the trigonometric functions in the formulas, we need to convert the angle from degrees to radians. The formula for this conversion is:
angle_in_radians = angle_in_degrees * (π / 180)
Let's apply this to our angle:
angle_in_radians = 23° * (π / 180) ≈ 0.4014 radians (rounded to four decimal places).
Step 2: Substitute the values into the formulas.
Using the formulas for x and y, we can calculate the rectangular coordinates:
x = -83 * cos(0.4014)
y = -83 * sin(0.4014)
Calculating the trigonometric functions for the angle in radians, we get:
cos(0.4014) ≈ 0.9179 (rounded to four decimal places).
sin(0.4014) ≈ 0.3979 (rounded to four decimal places).
Now we can substitute these values into the formulas:
x = -83 * 0.9179 ≈ -76.05 (rounded to the nearest hundredth).
y = -83 * 0.3979 ≈ -33.01 (rounded to the nearest hundredth).
Therefore, the rectangular coordinates for the given polar coordinates (-83, 23°) are approximately (-76.05, -33.01) when rounded to the nearest hundredth.