Find the corresponding rectangular coordinates for the polar point (4, -/6).
Is that supposed to be (4, -ƒÎ/6)?
If so, you would use this formula:
x = rcosӨ
y = rsinӨ
x = 4cos(-ƒÎ/6)
y = 4sin(-ƒÎ/6)
-ƒÎ/6 on the unit circle is the same as 11ƒÎ/6. The cosine (or x-value) for this angle is �ã3/2, and the sine (or y-value) for this angle is -1/2.
x = 4(�ã3/2)
y = 4(-1/2)
x = 2�ã3
y = -2
Your answer is:
(2�ã3, -2)
To find the corresponding rectangular coordinates for a given polar point, you need to use the following relationships:
x = r * cos(theta)
y = r * sin(theta)
In this case, the polar point is (4, -π/6). Plugging the values into the formulas:
x = 4 * cos(-π/6)
y = 4 * sin(-π/6)
To evaluate these trigonometric functions, let's recall the unit circle. The point (-π/6) corresponds to an angle of -30 degrees or 330 degrees, which lies in the fourth quadrant. In the unit circle, cos(-30°) and sin(-30°) are equal to cos(330°) and sin(330°) respectively.
So, cos(-π/6) = cos(-30°) = cos(330°) = √3/2
and sin(-π/6) = sin(-30°) = sin(330°) = -1/2
Now we can substitute these values into our formulas:
x = 4 * (√3/2) = 2√3
y = 4 * (-1/2) = -2
Hence, the corresponding rectangular coordinates for the polar point (4, -π/6) are (2√3, -2).