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:

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

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