Approximation of tangent fx

What is the non-trig formula (not a polynomial fit) for the growth curve that solves algebraically for the increase between tan(1 degree), tan( 2 degrees) continuing up to tangent(45 degrees)? okay to use pi Check calculation for 12°

Correct answer:

y =  0.2619

Step-by-step explanation:

Δ=tan1°=tanπ/180=0.01746 tan45°=1 α=12 °  β=α45=1245=33 ° y=1Δ β1+Δ β  y=Δ+(α1)/45=0.0175+(121)/450.2619  Verifying Solution:   y2=tanα°=tan12° =0.212557=0.21256  t1=tan1°=tanπ/180=0.01746 t2=tan2°=tanπ/90=0.03492 Δ2=t2t1=0.03490.01750.0175 t3=tan3°=tanπ/60=0.05241 Δ3=t3t2=0.05240.03490.0175 t4=tan4°=tanπ/45=0.06993 Δ4=t4t3=0.06990.05240.0175 t5=tan5°=tanπ/36=0.08749 Δ5=t5t4=0.08750.06990.0176 t6=tan6°=tanπ/30=0.1051 Δ6=t6t5=0.10510.08750.0176 t7=tan7°=tan7π/180=0.12278 Δ7=t7t6=0.12280.10510.0177 t8=tan8°=tan2π/45=0.14054 Δ8=t8t7=0.14050.12280.0178 t9=tan9°=tanπ/20=0.15838 Δ9=t9t8=0.15840.14050.0178 t10=tan10°=tanπ/18=0.17633 Δ10=t10t9=0.17630.15840.0179 t11=tan11°=tan11π/180=0.19438 Δ11=t11t10=0.19440.17630.0181 t12=tan12°=tanπ/15=0.21256 Δ12=t12t11=0.21260.19440.0182 t13=tan13°=tan13π/180=0.23087 Δ13=t13t12=0.23090.21260.0183 t14=tan14°=tan7π/90=0.24933 Δ14=t14t13=0.24930.23090.0185 t15=tan15°=tanπ/12=0.26795 Δ15=t15t14=0.26790.24930.0186 t16=tan16°=tan4π/45=0.28675 Δ16=t16t15=0.28670.26790.0188 t17=tan17°=tan17π/180=0.30573 Δ17=t17t16=0.30570.28670.019 t18=tan18°=tanπ/10=0.32492 Δ18=t18t17=0.32490.30570.0192 t19=tan19°=tan19π/180=0.34433 Δ19=t19t18=0.34430.32490.0194 t20=tan20°=tanπ/9=0.36397 Δ20=t20t19=0.3640.34430.0196 t21=tan21°=tan7π/60=0.38386 Δ21=t21t20=0.38390.3640.0199 t22=tan22°=tan11π/90=0.40403 Δ22=t22t21=0.4040.38390.0202 t23=tan23°=tan23π/180=0.42447 Δ23=t23t22=0.42450.4040.0204 t24=tan24°=tan2π/15=0.44523 Δ24=t24t23=0.44520.42450.0208 t25=tan25°=tan5π/36=0.46631 Δ25=t25t24=0.46630.44520.0211 t26=tan26°=tan13π/90=0.48773 Δ26=t26t25=0.48770.46630.0214 t27=tan27°=tan3π/20=0.50953 Δ27=t27t26=0.50950.48770.0218 t28=tan28°=tan7π/45=0.53171 Δ28=t28t27=0.53170.50950.0222 t29=tan29°=tan29π/180=0.55431 Δ29=t29t28=0.55430.53170.0226 t30=tan30°=tanπ/6=0.57735 Δ30=t30t29=0.57740.55430.023 t31=tan31°=tan31π/180=0.60086 Δ31=t31t30=0.60090.57740.0235 t32=tan32°=tan8π/45=0.62487 Δ32=t32t31=0.62490.60090.024 t33=tan33°=tan11π/60=0.64941 Δ33=t33t32=0.64940.62490.0245 t34=tan34°=tan17π/90=0.67451 Δ34=t34t33=0.67450.64940.0251 t35=tan35°=tan7π/36=0.70021 Δ35=t35t34=0.70020.67450.0257 t36=tan36°=tanπ/5=0.72654 Δ36=t36t35=0.72650.70020.0263 t37=tan37°=tan37π/180=0.75355 Δ37=t37t36=0.75360.72650.027 t38=tan38°=tan19π/90=0.78129 Δ38=t38t37=0.78130.75360.0277 t39=tan39°=tan13π/60=0.80978=1403π/5443 Δ39=t39t38=0.80980.78130.0285 t40=tan40°=tan2π/9=0.8391 Δ40=t40t39=0.83910.80980.0293 t41=tan41°=tan41π/180=0.86929 Δ41=t41t40=0.86930.83910.0302 t42=tan42°=tan7π/30=0.9004 Δ42=t42t41=0.90040.86930.0311 t43=tan43°=tan43π/180=0.93252 Δ43=t43t42=0.93250.90040.0321 t44=tan44°=tan11π/45=0.96569 Δ44=t44t43=0.96570.93250.0332 t45=tan45°=tanπ/4=1 Δ45=t45t44=10.96570.0343

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