Respuesta :
Answer:
a) [tex]\left(x,y\right)=\left(4.95,-4.95\right)[/tex]
b) [tex]r\angle\theta = 7\angle0.5236\,\text{radians}[/tex]
Step-by-step explanation:
Polar coordinates are represented as: [tex]r\angle\theta[/tex], where 'r' is the length (or magnitude) of the line, and '[tex]\theta[/tex]' is the angle measured from the positive x-axis.
in our case:
[tex]7\angle\dfrac{3\pi}{4}[/tex]
to covert the polar to cartesian:
[tex]x = r\cos{\theta}[/tex]
[tex]y = r\sin{\theta}[/tex]
we can plug in our values:
[tex]x = 7\cos{\dfrac{3\pi}{4}} = -7\dfrac{\sqrt{2}}{2}[/tex]
[tex]y = 7\sin{\dfrac{3\pi}{4}} = 7\dfrac{\sqrt{2}}{2}[/tex]
the Cartesian coordinates are:
[tex]\left(x,y\right)=\left(-7\dfrac{\sqrt{2}}{2},7\dfrac{\sqrt{2}}{2}\right)[/tex]
[tex]\left(x,y\right)=\left(4.95,-4.95\right)[/tex]
(b) to convert (x,y) = (6.06,-3.5)
we'll use the pythagoras theorem to find 'r'
[tex]r^2 = x^2+y^2[/tex]
[tex]r^2 = (6.06)^2+(-3.5)^2[/tex]
[tex]r = \sqrt{48.97} \approx 7[/tex]
the angle can be found by:
[tex]\tan{\theta} = \dfrac{y}{x}[/tex]
[tex]\tan{\theta} = \dfrac{3.5}{6.06}[/tex]
[tex]\theta = \arctan{left(\dfrac{3.5}{6.06}\right)}[/tex]
[tex]\theta = 0.5236 \text{radians}[/tex]
to convert radians to degrees:
[tex]\theta = 0.5236 \times \dfrac{180}{\pi} \approx 30^\circ[/tex]
writing in polar coordinates:
[tex]r\angle\theta = 7\angle30^\circ\,\,\text{OR}\,\,7\angle0.5236\,\text{radians}[/tex]
