What is the 3 dB beamwidth of the antenna radiation pattern shown in Figure E9-1?
Why The 3 dB beamwidth is the total angular width of the main lobe measured between the two points where the radiated power has fallen to half (3 dB below) the peak value. On Figure E9-1 you locate the peak of the major lobe, step in one ring to the 3 dB down circle, and read the bearing where the pattern crosses it on each side of the peak. Those crossings are about 25 degrees either side of the lobe axis, so the full beamwidth is 50 degrees.
Watch out The choice saying 25 degrees is the half-beamwidth, the angle from the lobe center out to one 3 dB point, not the width between both points; 75 degrees is closer to where the pattern has dropped much further down, near the lobe edges or nulls.
Beamwidth spans both sides of the peak: find one 3 dB point, then double the angle.