Can you check my work, please? This is for a trig study guide for my final next week. I'd like to know if I'm doing this right. :)
2. The polar coordinates of a point are given. Find the rectangular coordinates of each point.
a. (5,(π / 4))
b. (-2,(π / 6))
Answer:
a. x = M cos θ
x = 5(cos π/4)
x = 5(0.707)
x = 3.535
y = M sin θ
y = 5(sin π/4)
y = 5(0.707)
y = 3.535
(5,(π / 4)) = (3.535, 3.535)
b. x = M cos θ
x = -2(cos π/6)
x = -2(0.866)
x = -1.732
y = M sin θ
y = -2(sin π/6)
y = -2(0.5)
y = -1
(-2,(π / 6)) = (-1.732, -1)
both are correct
Thanks, Steve! :)I appreciate your help!
To check your work, we can convert the rectangular coordinates back to polar coordinates and see if they match the original given polar coordinates.
For (a) (5, π/4):
To convert from rectangular coordinates to polar coordinates, we use the formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
For the given rectangular coordinates (3.535, 3.535):
r = √((3.535)^2 + (3.535)^2) ≈ 4.98 (rounding to two decimal places)
θ = arctan(3.535/3.535) = π/4 (which is the same as the given polar angle)
The converted polar coordinates are approximately (4.98, π/4), which matches the original polar coordinates (5, π/4). So your answer for part (a) is correct.
For (b) (-2, π/6):
Again, to convert from rectangular coordinates to polar coordinates, we use the formulas:
r = √(x^2 + y^2)
θ = arctan(y/x)
For the given rectangular coordinates (-1.732, -1):
r = √((-1.732)^2 + (-1)^2) ≈ 2 (rounding to two decimal places)
θ = arctan(-1/-1.732) ≈ π/6 (rounding to two decimal places)
The converted polar coordinates are approximately (2, π/6), which matches the original polar coordinates (-2, π/6). So your answer for part (b) is also correct.
Overall, your work is correct for both parts (a) and (b). Keep up the good work!