QQuestionMathematics
QuestionMathematics
The Cartesian coordinates of a point are given.
(a) (โ6, 6)
(i) Find polar coordinates (r, ๐) of the point, where r > 0 and 0 โค ๐ < 2๐.
(ii) Find polar coordinates (r, ๐) of the point, where r < 0 and 0 โค ๐ < 2๐.
(b) (4, 4)
(i)Find polar coordinates (r, ๐) of the point, where r > 0 and 0 โค ๐ < 2๐.
(ii) Find polar coordinates (r, ๐) of the point, where r < 0 and 0 โค ๐ < 2๐.
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Answer
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Step 1:
So, the polar coordinates of the point are $(6\sqrt{2}, \frac{7\pi}{4})$.
Substituting these values into the formulas, we get:
Step 2:
So, the polar coordinates of the point are $(-6\sqrt{2}, \frac{15\pi}{4})$.
In this case, we need to add $\pi$ to our value of $\theta$:
Step 3:
Since $0 \leq \theta < 2\pi$, we don't need to add any multiples of $2\pi$ to our result.
Using the same formulas as before, we get:
Step 4:
So, the polar coordinates of the point are $(-4\sqrt{2}, \frac{5\pi}{4})$.
In this case, we need to add $\pi$ to our value of $\theta$:
Final Answer
(a) (i) $(6\sqrt{2}, \frac{7\pi}{4})$ (a) (ii) $(- 6\sqrt{2}, \frac{15\pi}{4})$ (b) (i) $(4\sqrt{2}, \frac{\pi}{4})$ (b) (ii) $(- 4\sqrt{2}, \frac{5\pi}{4})$
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