Answer to Question #273950 in Analytic Geometry for Aliyah

Question

1. Convert the polar coordinates (-8, 2π/3) into rectangular coordinates.
2. Convert the rectangular coordinates (3, -3) into polar coordinates with r > 0 and 0 ≤ θ < 2π.
3. Convert the rectangular equation x2 + y2 = 100 into a polar equation that expresses r in terms of θ.
4. Convert the polar equation 4r cos θ + r sin θ = 8 into a rectangular equation that expresses y in terms of x.

Answer 1

Let $(r, \theta)$ - polar coordinates and $(x, y)$ - rectangular coordinates.

1) $r_{1} = - 8$

$\theta_{1} = \dfrac{2\pi}{3}$

$x_{1} = r_{1}cos(\theta_{1}) = (-8) * cos \left(\dfrac{2\pi}{3}\right) = (-8) * \left(-\dfrac{1}{2}\right) = 4$

$y_{1} = r_{1}sin(\theta_{1}) = (-8) * sin \left(\dfrac{2\pi}{3}\right) = (-8) * \left(\dfrac{\sqrt{3}}{2}\right) = -4\sqrt{3}$

2) $x_{2} = 3$

$y_{2} = - 3$

$r_{2} > 0$

$0 \le \theta_{2} < 2\pi$

$r_{2}^{2} = x_{2}^{2} + y_{2}^{2} = 3^{2} + (-3)^{2} = 18 \rArr r_{2} = 3\sqrt{2}$

$tan(\theta_{2}) = \dfrac{y_{2}}{x_{2}} = -1$

$x_{2} > 0, y_{2} < 0 \rArr \theta_{2} = \dfrac{7\pi}{4}$

3)$x^{2} + y^{2} = 100$

$r^{2} = 100 \rArr r = 10$

4) $4 r cos\theta + rsin \theta = 8$

$4x + y = 8$

$y = - 4x + 8$