Care for your voice. Avoid behaviors that compromise the strength of your voice, such as shouting or screaming. Take care of your general health.
Warm up as instructed by your vocal coach. Ask if he would like you to warm up before lessons.
Record yourself singing a variety of vowel sounds throughout your vocal range. Experiment with different consonants, as this will change the way you approach the vowel sound. Sing slowly, so you can hear the difference clearly.
Watch the video. Note the times on the recording when your voice seems to ring. Note which pitches and vowel sounds you are singing when your voice has this ring. Note times, pitches and vowel sounds that seem to be especially weak.
Watch the video with the sound off. Note your posture and embouchure -- the position of your mouth -- during times when your voice seems to ring. Examine your posture and embouchure when your tone seems to lack resonance.
Work the especially resonant points into your singing. This may require altering vowel sounds slightly, singing in certain keys to take advantage of the resonance offered by certain notes or adjusting your posture and/or embouchure. The proper approach for each piece will be determined by the requirements of the work you are singing.