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Find all points of intersections for the polar curves r = \sin(2 \theta) and r= \cos(2 \theta).

Question:

Find all points of intersections for the polar curves {eq}r = \sin(2 \theta) {/eq} and {eq}r= \cos(2 \theta) {/eq}.

Coterminal angles:

Two or more angles are called coterminals, when they have the same initial side and the same final side.The difference between two or more coterminal angles is the number of turns on the initial side.

Answer and Explanation:

Find all points of intersections for the polar curves.

{eq}r = \sin(2 \theta)\\ r= \cos(2 \theta) \\ \sin(2 \theta)= \cos(2 \theta) {/eq}.

The sine and the cosine they have the same sign in the first and third quadrate

{eq}\sin(2 \theta)= \cos(2 \theta)\\ 2\theta_1 = \frac {\pi}{4} +2k\pi \,\, k \,\, \in \,\, Z\\ \theta_1= \frac {\pi}{8} +k\pi \,\, k \,\, \in \,\, Z\\ 2\theta_2 = \frac {5\pi}{4} +2k\pi \,\, k \,\, \in \,\, Z\\ \theta_1= \frac {5\pi}{8} +k\pi \,\, k \,\, \in \,\, Z {/eq}.

The points of intersections for the polar curves {eq}r = \sin(2 \theta) {/eq} and {eq}r= \cos(2 \theta) {/eq} are:

{eq}S= \{ \frac {\pi}{8} +k\pi, \frac {5\pi}{8}+k\pi : \,\, k \,\, \in \,\, Z \} {/eq}


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