  We present a study of the detection efficiency for the TESS mission, focusing
on the yield of longer-period transiting exoplanets ($P > 25$ days). We created
the Transit Investigation and Recoverability Application (TIaRA) pipeline to
use real TESS data with injected transits to create sensitivity maps which we
combine with occurrence rates derived from Kepler. This allows us to predict
longer-period exoplanet yields, which will help design follow-up photometric
and spectroscopic programs, such as the NGTS Monotransit Program. For the TESS
Year 1 and Year 3 SPOC FFI lightucurves, we find $2271^{+241}_{-138}$
exoplanets should be detectable around AFGKM dwarf host stars. We find
$215^{+37}_{-23}$ exoplanets should be detected from single-transit events or
"monotransits". An additional $113^{+22}_{-13}$ detections should result from
"biennial duotransit" events with one transit in Year 1 and a second in Year 3.
We also find that K dwarf stars yield the most detections by TESS per star
observed. When comparing our results to the TOI catalogue we find our
predictions agree within $1\sigma$ of the number of discovered systems with
periods between 0.78 and 6.25 days and agree to $2\sigma$ for periods between
6.25 and 2 days. Beyond periods of 25 days we predict $403^{+64}_{-38}$
detections, which is 3 times as many detections as there are in the TOI
catalogue with $>3\sigma$ confidence. This indicates a significant number of
long-period planets yet to be discovered from TESS data as monotransits or
biennial duotransits.
