while the rectangular coordinates show the position of a point along the xx or yy axis, polar coordinates show the distance of the point from (0,0)(0,0) and the angle that the point makes with the xx-axis.
the rectangular coordinates here are (-sqrt2, sqrt2)(−√2,√2). if a horizontal line was drawn, then a vertical line, the horizontal line would extend to a point sqrt2√2 to the left of (0,0)(0,0) and the vertical line would extend from this new point to the point sqrt2√2 from the left of (0,0)(0,0) and sqrt2√2 above (0,0)(0,0).
however, this can be drawn as one line. the ends of this line are at the ends of the lines where they do not meet. this forms a triangle with three sides that meet at three vertices.
since the Cartesian graph is right-angled, the triangle is also right-angled.
this means that we can use Pythagoras' theorem to solve for the length of the new line, which is the hypotenuse.
a^2 + b^2 = c^2a2+b2=c2
aa is the distance along the xx-axis, which is -sqrt2−√2.
bb is the distance along the yy-axis, which is sqrt2√2
cc is the length of the new line.
(-sqrt2)^2 + (sqrt2)^2 = 2 + 2 = 4(−√2)2+(√2)2=2+2=4
c^2 = 4c2=4
c = +-2c=±2
since the point is to the left of (0,0)(0,0), the line is also travelling left of (0,0)(0,0). this means that the length of line cc is negative.
c = -2c=−2, so the distance of the point from (0,0)(0,0) is -2−2.
meanwhile, we can again use two sides of the triangle to find the size of the angle between the line cc and the xx-axis. the angle can be labelled thetaθ.
the xx-length is -sqrt2−√2. this is the adjacent to the angle.
the yy-length is sqrt2√2. this is the opposite to the angle.
tan theta = O/Atanθ=∅A
here, tan theta = (sqrt2)/(-sqrt2)tanθ=√2−√2, which is -1−1.
then the inverse tan function could be applied to find theta:θ:
theta = tan^-1(-1) = -45^@θ=tan−1(−1)=−45∘.
the coordinates are (-2, 45^@)(−2,45∘)
