Determine as coordenadas retangulares dos pontos (lembrete: x=r \cos (\theta) e y=r \operatorname{sen}(\theta) ):
P_{3}\left(-6, \frac{-5 \pi}{6}\right)

Passo 1

O enunciado pede para converter as coordenadas polares P_{3}(r,\theta)=\bigl(-6,\,-\tfrac{5\pi}{6}\bigr) em coordenadas retangulares, usando as fórmulas
x = r\cos(\theta),
y = r\operatorname{sen}(\theta).

Passo 2

Calculamos os valores de \cos(\theta) e \operatorname{sen}(\theta) para \theta = -\tfrac{5\pi}{6}: \cos\bigl(-\tfrac{5\pi}{6}\bigr) = \cos\bigl(\tfrac{5\pi}{6}\bigr) = -\tfrac{\sqrt{3}}{2}, \operatorname{sen}\bigl(-\tfrac{5\pi}{6}\bigr) = -\operatorname{sen}\bigl(\tfrac{5\pi}{6}\bigr) = -\tfrac{1}{2}.

Passo 3

Aplicamos as fórmulas de conversão com r = -6:
x = (-6)\cdot\Bigl(-\tfrac{\sqrt{3}}{2}\Bigr),
y = (-6)\cdot\Bigl(-\tfrac{1}{2}\Bigr).

Passo 4

Fazendo as contas:
x = 3\sqrt{3},
y = 3.

Resposta

As coordenadas retangulares de P_{3} são
\bigl(3\sqrt{3},\,3\bigr).

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