AN SMA TWIN INSTRUCTOR PLT HAD A BADLY PRECESSING HDG GYROSCOPE IN IMC.

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WHILE TAXIING; I PERFORMED AN INST CHK AND EXPLAINED ITS SIGNIFICANCE TO MY STUDENT/PAX. I WAS CLRED TO POS AND HOLD (LINE-UP) ON RWY 34A. AFTER LINING UP; I CHKED HDG INDICATOR; ALT INDICATOR AND TURNED ON XPONDER; LIGHTS AND STARTED TIMER. ABOUT A MIN LATER; I WAS CLRED FOR TKOF. SHORTLY AFTER TKOF; I WAS GIVEN A VECTOR TO TURN L TO 280 DEGS. I WAS NOW IN AND OUT OF IMC. BEFORE STARTING L I TOOK A BRIEF LOOK AT MY HDG INDICATOR AND IT SHOWED 280 DEGS AT MY 2-3 O'CLOCK POS. OBSERVING THIS; I CALLED TWR AND QUESTIONED THEIR INSTRUCTIONS. TWR CONFIRMED L TO 280 DEGS AND SAID TO EXPEDITE IMMEDIATELY THEN HANDED ME OFF TO DEP. I RELUCTANTLY STARTED A L TURN; THINKING PERHAPS THEY WANT ME TO PASS BACK OVER THE ARPT FOR SEPARATION OR TIMING. AS I ROLLED OUT ON THE HDG INDICATOR AT 280 DEGS; DEP ASKED ME WHAT I WAS 'DOING UP THERE.' I EXPLAINED THAT I WAS COMPLYING WITH TWR'S VECTORS. DEP ADVISED ME I WAS NOWHERE NEAR 280 DEGS AND GAVE MY 'NO-GYROSCOPE' VECTORS BACK TO COURSE. A QUICK CHK OF MY COMPASS AND HDG INDICATOR SHOWED THAT; IN FACT; MY HDG INDICATOR HAD TUMBLED SOME 90 DEGS PLUS OR MINUS; OR SO. I INFORMED ATC OF THE SIT AND RESET THE HDG INDICATOR TO THE PROPER HDG. THE REST OF THE FLT WAS UNEVENTFUL; HOWEVER; THERE WAS SIGNIFICANT PRECESSION OF THE HDG INDICATOR THROUGHOUT. IN THE PAST; I; AND OTHER PLTS WHO FLEW THIS ACFT; HAD WRITTEN UP PROBS WITH THE INST IN QUESTION. BEFORE TAKING THE AIRPLANE I NOTICED THAT THE DISCREPANCY SHEET WAS CLR AND 'ASSUMED' THE PROBS HAD BEEN CORRECTED. UNFORTUNATELY; IT WAS NOT 'CONSIDERED' A SEVERE 'WRITE-UP' AS IT WAS A TRAINING ACFT AND IT ACTED UP INTERMITTENTLY.