El gráfico de la función cruce el eje A con f = 0
o sea hay que resolver la ecuación:
$$\sin{\left(4 a \right)} + \cos{\left(a \right)} = 0$$
Resolvermos esta ecuaciónPuntos de cruce con el eje A:
Solución analítica$$a_{1} = - \frac{5 \pi}{6}$$
$$a_{2} = - \frac{\pi}{2}$$
$$a_{3} = - \frac{\pi}{6}$$
$$a_{4} = - \frac{\pi}{10}$$
$$a_{5} = \frac{\pi}{2}$$
$$a_{6} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
$$a_{7} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} + \frac{i}{4} \right)}$$
$$a_{8} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}$$
Solución numérica$$a_{1} = -59.3761011528471$$
$$a_{2} = 7.85398163397448$$
$$a_{3} = 93.9336203423348$$
$$a_{4} = -0.314159265358979$$
$$a_{5} = -25.6563400043166$$
$$a_{6} = -17.9070781254618$$
$$a_{7} = -69.6386371545737$$
$$a_{8} = -65.6592864600267$$
$$a_{9} = -85.7654794430014$$
$$a_{10} = -69.4291976443344$$
$$a_{11} = -34.0339204138894$$
$$a_{12} = -38.0132711084365$$
$$a_{13} = -84.2994028713261$$
$$a_{14} = 95.8185759344887$$
$$a_{15} = 83.8805238508475$$
$$a_{16} = 87.6504350351552$$
$$a_{17} = -45.553093477052$$
$$a_{18} = -40.317105721069$$
$$a_{19} = -99.5884871187965$$
$$a_{20} = 90.1637091580271$$
$$a_{21} = -98.9601685880785$$
$$a_{22} = -58.1194640914112$$
$$a_{23} = 70.0575161750524$$
$$a_{24} = 97.9129710368819$$
$$a_{25} = -55.6061899685393$$
$$a_{26} = 43.6681378848981$$
$$a_{27} = 60.2138591938044$$
$$a_{28} = 2.19911485751286$$
$$a_{29} = -80.1106126665397$$
$$a_{30} = 86.3937979737193$$
$$a_{31} = 46.18141200777$$
$$a_{32} = 53.9306738866248$$
$$a_{33} = -43.0398193541802$$
$$a_{34} = 91.420346219463$$
$$a_{35} = 76.9690200129499$$
$$a_{36} = 16.2315620435473$$
$$a_{37} = -14.1371669411541$$
$$a_{38} = -54.9778714378214$$
$$a_{39} = -73.8274273593601$$
$$a_{40} = 5.96902604182061$$
$$a_{41} = 62.3082542961976$$
$$a_{42} = -61.8893752757189$$
$$a_{43} = 76.340701482232$$
$$a_{44} = -21.6769893097696$$
$$a_{45} = -29.845130209103$$
$$a_{46} = -89.5353906273091$$
$$a_{47} = 64.4026493985908$$
$$a_{48} = 32.3584043319749$$
$$a_{49} = -48.0663675999238$$
$$a_{50} = 12.2522113490002$$
$$a_{51} = 18.3259571459405$$
$$a_{52} = 29.845130209103$$
$$a_{53} = -51.8362787842316$$
$$a_{54} = 28.5884931476671$$
$$a_{55} = -81.9955682586936$$
$$a_{56} = -21.4675497995303$$
$$a_{57} = -88.2787535658732$$
$$a_{58} = -36.1283155162826$$
$$a_{59} = -7.85398163397448$$
$$a_{60} = 14.1371669411541$$
$$a_{61} = 504.225620901162$$
$$a_{62} = -92.0486647501809$$
$$a_{63} = 80.1106126665397$$
$$a_{64} = 26.0752190247953$$
$$a_{65} = -27.7507351067098$$
$$a_{66} = 51.2079602535136$$
$$a_{67} = 42.4115008234622$$
$$a_{68} = -44.2964564156161$$
$$a_{69} = 22.3053078404875$$
$$a_{70} = 100.007366139275$$
$$a_{71} = 66.2876049907446$$
$$a_{72} = 49.9513231920777$$
$$a_{73} = -31.7300858012569$$
$$a_{74} = 36.1283155162826$$
$$a_{75} = -78.0162175641465$$
$$a_{76} = 56.2345084992573$$
$$a_{77} = -1.5707963267949$$
$$a_{78} = -95.8185759344887$$
$$a_{79} = -41.7831822927443$$
$$a_{80} = -4.08407044966673$$
$$a_{81} = -6.80678408277789$$
$$a_{82} = 72.5707902979242$$
$$a_{83} = -71.733032256967$$
$$a_{84} = -15.39380400259$$
$$a_{85} = 39.8982267005904$$
$$a_{86} = 9.94837673636768$$
$$a_{87} = 47.4380490692059$$
$$a_{88} = 4869.15445379882$$
$$a_{89} = -4.71238898038469$$
$$a_{90} = -75.712382951514$$
$$a_{91} = 24.8185819633594$$
$$a_{92} = 3.45575191894877$$
$$a_{93} = 68.8008791136165$$
$$a_{94} = 73.8274273593601$$
$$a_{95} = -11.6238928182822$$
$$a_{96} = 58.1194640914112$$