Approximation of tangent fx

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

Correct answer:

y =  0.2619

Step-by-step explanation:

$$\begin{gathered} \Delta =\tan 1°=\tan \pi /180=0.01746 \\ \tan 45°=1 \\ \alpha =12{}^{\circ } \\ \beta =|\alpha -45|=|12-45|=33{}^{\circ } \\ y=\frac{1-\Delta \cdot \beta }{1+\Delta \cdot \beta } \\ y=\Delta +(\alpha -1)/45=0.0175+(12-1)/45\doteq 0.2619 \\ \text{ Verifying Solution: } \\ {y}_2=\tan \alpha =\tan 12°=0.212557=0.21256 \\ {t}_1=\tan 1°=\tan \pi /180=0.01746 \\ {t}_2=\tan 2°=\tan \pi /90=0.03492 \\ {\Delta }_2=t_2-t_1=0.0349-0.0175\doteq 0.0175 \\ {t}_3=\tan 3°=\tan \pi /60=0.05241 \\ {\Delta }_3=t_3-t_2=0.0524-0.0349\doteq 0.0175 \\ {t}_4=\tan 4°=\tan \pi /45=0.06993 \\ {\Delta }_4=t_4-t_3=0.0699-0.0524\doteq 0.0175 \\ {t}_5=\tan 5°=\tan \pi /36=0.08749 \\ {\Delta }_5=t_5-t_4=0.0875-0.0699\doteq 0.0176 \\ {t}_6=\tan 6°=\tan \pi /30=0.1051 \\ {\Delta }_6=t_6-t_5=0.1051-0.0875\doteq 0.0176 \\ {t}_7=\tan 7°=\tan 7\pi /180=0.12278 \\ {\Delta }_7=t_7-t_6=0.1228-0.1051\doteq 0.0177 \\ {t}_8=\tan 8°=\tan 2\pi /45=0.14054 \\ {\Delta }_8=t_8-t_7=0.1405-0.1228\doteq 0.0178 \\ {t}_9=\tan 9°=\tan \pi /20=0.15838 \\ {\Delta }_9=t_9-t_8=0.1584-0.1405\doteq 0.0178 \\ {t}_{10}=\tan 10°=\tan \pi /18=0.17633 \\ {\Delta }_{10}=t_{10}-t_9=0.1763-0.1584\doteq 0.0179 \\ {t}_{11}=\tan 11°=\tan 11\pi /180=0.19438 \\ {\Delta }_{11}=t_{11}-t_{10}=0.1944-0.1763\doteq 0.0181 \\ {t}_{12}=\tan 12°=\tan \pi /15=0.21256 \\ {\Delta }_{12}=t_{12}-t_{11}=0.2126-0.1944\doteq 0.0182 \\ {t}_{13}=\tan 13°=\tan 13\pi /180=0.23087 \\ {\Delta }_{13}=t_{13}-t_{12}=0.2309-0.2126\doteq 0.0183 \\ {t}_{14}=\tan 14°=\tan 7\pi /90=0.24933 \\ {\Delta }_{14}=t_{14}-t_{13}=0.2493-0.2309\doteq 0.0185 \\ {t}_{15}=\tan 15°=\tan \pi /12=0.26795 \\ {\Delta }_{15}=t_{15}-t_{14}=0.2679-0.2493\doteq 0.0186 \\ {t}_{16}=\tan 16°=\tan 4\pi /45=0.28675 \\ {\Delta }_{16}=t_{16}-t_{15}=0.2867-0.2679\doteq 0.0188 \\ {t}_{17}=\tan 17°=\tan 17\pi /180=0.30573 \\ {\Delta }_{17}=t_{17}-t_{16}=0.3057-0.2867\doteq 0.019 \\ {t}_{18}=\tan 18°=\tan \pi /10=0.32492 \\ {\Delta }_{18}=t_{18}-t_{17}=0.3249-0.3057\doteq 0.0192 \\ {t}_{19}=\tan 19°=\tan 19\pi /180=0.34433 \\ {\Delta }_{19}=t_{19}-t_{18}=0.3443-0.3249\doteq 0.0194 \\ {t}_{20}=\tan 20°=\tan \pi /9=0.36397 \\ {\Delta }_{20}=t_{20}-t_{19}=0.364-0.3443\doteq 0.0196 \\ {t}_{21}=\tan 21°=\tan 7\pi /60=0.38386 \\ {\Delta }_{21}=t_{21}-t_{20}=0.3839-0.364\doteq 0.0199 \\ {t}_{22}=\tan 22°=\tan 11\pi /90=0.40403 \\ {\Delta }_{22}=t_{22}-t_{21}=0.404-0.3839\doteq 0.0202 \\ {t}_{23}=\tan 23°=\tan 23\pi /180=0.42447 \\ {\Delta }_{23}=t_{23}-t_{22}=0.4245-0.404\doteq 0.0204 \\ {t}_{24}=\tan 24°=\tan 2\pi /15=0.44523 \\ {\Delta }_{24}=t_{24}-t_{23}=0.4452-0.4245\doteq 0.0208 \\ {t}_{25}=\tan 25°=\tan 5\pi /36=0.46631 \\ {\Delta }_{25}=t_{25}-t_{24}=0.4663-0.4452\doteq 0.0211 \\ {t}_{26}=\tan 26°=\tan 13\pi /90=0.48773 \\ {\Delta }_{26}=t_{26}-t_{25}=0.4877-0.4663\doteq 0.0214 \\ {t}_{27}=\tan 27°=\tan 3\pi /20=0.50953 \\ {\Delta }_{27}=t_{27}-t_{26}=0.5095-0.4877\doteq 0.0218 \\ {t}_{28}=\tan 28°=\tan 7\pi /45=0.53171 \\ {\Delta }_{28}=t_{28}-t_{27}=0.5317-0.5095\doteq 0.0222 \\ {t}_{29}=\tan 29°=\tan 29\pi /180=0.55431 \\ {\Delta }_{29}=t_{29}-t_{28}=0.5543-0.5317\doteq 0.0226 \\ {t}_{30}=\tan 30°=\tan \pi /6=0.57735 \\ {\Delta }_{30}=t_{30}-t_{29}=0.5774-0.5543\doteq 0.023 \\ {t}_{31}=\tan 31°=\tan 31\pi /180=0.60086 \\ {\Delta }_{31}=t_{31}-t_{30}=0.6009-0.5774\doteq 0.0235 \\ {t}_{32}=\tan 32°=\tan 8\pi /45=0.62487 \\ {\Delta }_{32}=t_{32}-t_{31}=0.6249-0.6009\doteq 0.024 \\ {t}_{33}=\tan 33°=\tan 11\pi /60=0.64941 \\ {\Delta }_{33}=t_{33}-t_{32}=0.6494-0.6249\doteq 0.0245 \\ {t}_{34}=\tan 34°=\tan 17\pi /90=0.67451 \\ {\Delta }_{34}=t_{34}-t_{33}=0.6745-0.6494\doteq 0.0251 \\ {t}_{35}=\tan 35°=\tan 7\pi /36=0.70021 \\ {\Delta }_{35}=t_{35}-t_{34}=0.7002-0.6745\doteq 0.0257 \\ {t}_{36}=\tan 36°=\tan \pi /5=0.72654 \\ {\Delta }_{36}=t_{36}-t_{35}=0.7265-0.7002\doteq 0.0263 \\ {t}_{37}=\tan 37°=\tan 37\pi /180=0.75355 \\ {\Delta }_{37}=t_{37}-t_{36}=0.7536-0.7265\doteq 0.027 \\ {t}_{38}=\tan 38°=\tan 19\pi /90=0.78129 \\ {\Delta }_{38}=t_{38}-t_{37}=0.7813-0.7536\doteq 0.0277 \\ {t}_{39}=\tan 39°=\tan 13\pi /60=0.80978=1403\pi /5443 \\ {\Delta }_{39}=t_{39}-t_{38}=0.8098-0.7813\doteq 0.0285 \\ {t}_{40}=\tan 40°=\tan 2\pi /9=0.8391 \\ {\Delta }_{40}=t_{40}-t_{39}=0.8391-0.8098\doteq 0.0293 \\ {t}_{41}=\tan 41°=\tan 41\pi /180=0.86929 \\ {\Delta }_{41}=t_{41}-t_{40}=0.8693-0.8391\doteq 0.0302 \\ {t}_{42}=\tan 42°=\tan 7\pi /30=0.9004 \\ {\Delta }_{42}=t_{42}-t_{41}=0.9004-0.8693\doteq 0.0311 \\ {t}_{43}=\tan 43°=\tan 43\pi /180=0.93252 \\ {\Delta }_{43}=t_{43}-t_{42}=0.9325-0.9004\doteq 0.0321 \\ {t}_{44}=\tan 44°=\tan 11\pi /45=0.96569 \\ {\Delta }_{44}=t_{44}-t_{43}=0.9657-0.9325\doteq 0.0332 \\ {t}_{45}=\tan 45°=\tan \pi /4=1 \\ {\Delta }_{45}=t_{45}-t_{44}=1-0.9657\doteq 0.0343 \end{gathered}$$

Try another example