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tokens: afterglow emission takes place when the Lorentz factor of the core of the blast slows down to Γ ∼ 1/ θ v which falls into mildly relativistic domain if θ v is relatively large. Since we wish to follow the blast emission up to, and shortly after, its peak, a treatment that follows the blast deceleration down to a trans-relativistic speed is required. In this work we use a semi-analytical model following the shock jump conditions derived by an approximate trans-relativistic EoS (Uhm 2011 ). This model works well at both high and modest Lorentz factors and can smoothly transform to a Sedov–Taylor solution when the shock is no longer relativistic. Our model assumes one-dimensional radial solution for each blast segment. Therefore, we neglect sideways (lateral) expansion in our model. The sideways expansion of the blast is important only at later stages of the blast evolution and it affects the light curve predictions after its peak (Gill Granot 2018 ). However, the key factor to constrain inclination angle, of interest in this study, is the arising slope before the peak flux, where the jet is still relativistic. So the sideways expansion will not influence our result. To describe the physical quantities more clearly, here we consider two reference frames. One is burst rest frame (hereafter BR frame) which is at rest with the central engine. The other is the shock co-moving frame (hereafter CM frame) which is at rest with respect to the gas just downstream of the forward shock. Primed variables (i.e., denoted with ′) are measured in CM frame, while non-primed ones are measured in BR frame. Under the approximation of neglecting sideways expansion, we can regard the jet evolution at each radial direction as being independent. It is therefore convenient to describe the evolution at each direction by using isotropic equivalence quantities. Consider an isotropic burst of total energy E iso = Γ 0 M ej c 2 , where M ej is the mass of ejecta and Γ 0 is its initial Lorentz factor. The outflow spreads in a cold uniform environment of number density n and forms a relativistic shock. We assume that the blastwave is expanding adiabatically without further energy injection (for considerations of energy injection in the blast see Lamb et al. 2020 ). Energy loss due to radiation is negligible compared with E iso . We define the radius R of the shock by the distance from burst center to the forward shock. All swept-up mass is compressed in the downstream of forward shock, moving with Lorentz factor Γ measured in BR frame, and its isotropic kinetic energy is E k . We also assume that all ejecta are moving together with shocked ambient at the same Lorentz factor behind the contact discontinuity, and its total energy is E ej = Γ M ej c 2 . The trans-relativistic shock jump conditions is (Uhm 2011 ; Ryan et al. 2019 ) where , , and are the number density, internal energy, and pressure of shocked region, respectively. is the dimensionless speed. c is the speed of light.