Polar coordinates, matrices and transformations

The polar coordinates (r, θ) of a point P in a plane provide an alternative, and sometimes convenient, way of describing its position relative to a fixed point, the pole, in the plane; for example, ‘50 km from O, bearing 53·1°’ instead of ‘40 km east and 30 km north from O’. Taking the pole to coincide with the origin of a cartesian system, and the initial line Ol, from which θ is measured anti-clockwise, to coincide with the x-axis, Fig. 3.1 enables us to convert from cartesian to polar coordinates thus: (3.1)<math display='block'> <mrow> <mi>x</mi><mo>=</mo><mi>r</mi><mtext>&#x2009;</mtext><mi>cos</mi><mtext>&#x2009;</mtext><mi>&#x03B8;</mi><mo>,</mo><mtext>&#x2009;</mtext><mi>y</mi><mo>=</mo><mi>r</mi><mtext>&#x2009;</mtext><mi>sin</mi><mtext>&#x2009;</mtext><mi>&#x03B8;</mi><mo>.</mo> </mrow> </math> $$x = r\,\cos \,\theta ,\,y = r\,\sin \,\theta .$$