Question

For the polar point \(\displaystyle{\left(-{4},-{\frac{{{3}\pi}}{{{4}}}}\right)}\)
a) Give alternate coordinates with \(\displaystyle{r}{>}{0}\) and \(\displaystyle-{2}\pi\leq\theta{<}{0}\).
b) Give alternate coordinates with \(\displaystyle{r}{<}{0}\) and \(\displaystyle{0}\leq\theta{<}{2}\pi\).
c) Give alternate coordinates of your choice.

Answer 1

Step 1
Given: \(\displaystyle{P}{\left(-{4},-{\frac{{{3}\pi}}{{{4}}}}\right)}\)
a) \(\displaystyle{\left(-{4},-{\frac{{{3}\pi}}{{{4}}}}\right)}={\left({4},-{\frac{{{3}\pi}}{{{4}}}}+\pi\right)}\)
\(\displaystyle={\left({4},{\frac{{\pi}}{{{4}}}}\right)}\)
So, \(\displaystyle{\left(-{4},-{\frac{{{3}\pi}}{{{4}}}}\right)}={\left({4},{\frac{{\pi}}{{{4}}}}\right)}\)
Step 2
b) \(\displaystyle{\left(-{4},-{\frac{{{3}\pi}}{{{4}}}}\right)}={\left(-{4},-{\frac{{{3}\pi}}{{{4}}}}+{2}\pi\right)}\)
\(\displaystyle={\left(-{4},-{\frac{{{5}\pi}}{{{4}}}}\right)}\)
So, \(\displaystyle{\left(-{4},-{\frac{{{3}\pi}}{{{4}}}}\right)}={\left(-{4},{\frac{{{5}\pi}}{{{4}}}}\right)}\)
c) \(\displaystyle{\left(-{4},-{\frac{{{3}\pi}}{{{4}}}}\right)}={\left({4},-{\frac{{{3}\pi}}{{{4}}}}+{3}\pi\right)}\)
\(\displaystyle={\left({4},{\frac{{{9}\pi}}{{{4}}}}\right)}\)
So, \(\displaystyle{\left(-{4},-{\frac{{{3}\pi}}{{{4}}}}\right)}={\left({4},{\frac{{{9}\pi}}{{{4}}}}\right)}\)

Answer 2

Answer is given below (on video)