Hallar "x" en : sec(5x + 20º) = csc(x + 40º)?
Hallar "x" en : sec(5x + 20º) = csc(x + 40º).
Hallar "x" en : sec(5x + 20º) = csc(x + 40º).
En resumen
Antes de abordar el problema recordemos la relación que tienen las funciones SENO y COSENO <img src="https://tex.z-dn.net/?
Antes de abordar el problema recordemos la relación que tienen las funciones SENO y COSENO
<img src="https://tex.z-dn.net/?f=%5Csin%20%5Cleft%28x%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%5Cright%29%3D%5Csin%20x%5Ccos%20%5Cdfrac%7B%5Cpi%7D%7B2%7D%2B%5Ccos%20x%5Csin%20%5Cdfrac%7B%5Cpi%7D%7B2%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Csin%20%5Cleft%28x%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%5Cright%29%3D%5Ccos%20x%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BM%5C%27as%20all%5C%27a%3A%20%7D%5C%5C%20%5C%5C%20%0A%5Csin%20%5Cleft%5Bx%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%282k%2B1%29%5Cright%5D%3D%5Csin%20x%5Ccos%20%5Cleft%5B%5Cdfrac%7B%5Cpi%7D%7B2%7D%282k%2B1%29%5Cright%5D%2B%5Ccos%20x%5Csin%20%5Cleft%5B%5Cdfrac%7B%5Cpi%7D%7B2%7D%282k%2B1%29%5Cright%5D%5C%5C%20%5C%5C%20%5C%5C%0A%5Csin%20%5Cleft%5Bx%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%282k%2B1%29%5Cright%5D%3D%5Ccos%20x%5Csin%20%5Cleft%5B%5Cdfrac%7B%5Cpi%7D%7B2%7D%282k%2B1%29%5Cright%5D%0A" />
<img src="https://tex.z-dn.net/?f=%5Csin%20%5Cleft%5Bx%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%282k%2B1%29%5Cright%5D%3D%5Ccos%20x%5Csin%20%5Cleft%28%5Cpi%20k%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%5Cright%29%5C%5C%20%5C%5C%20%5C%5C%0A%5Csin%20%5Cleft%5Bx%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%282k%2B1%29%5Cright%5D%3D%5Ccos%20x%5Ccos%20%28%5Cpi%20k%29%5C%5C%20%5C%5C%20%5C%5C%0A%5Csin%20%5Cleft%5Bx%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%282k%2B1%29%5Cright%5D%3D%28-1%29%5Ek%5Ccos%20x%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctext%7BPor%20ende%20podemos%20garantizar%3A%20%7D%5C%5C%20%5C%5C%0A%5Csin%20%5Cleft%5Bx%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%284k%2B1%29%5Cright%5D%3D%5Ccos%20x%5C%2C%2C%5Cforall%20k%5Cin%20%5Cmathbb%7BZ%7D%20%5C%2C%5C%5C%20%5C%5C%20%0A%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%3D%0A%0A%0A" />
<img src="https://tex.z-dn.net/?f=%5Ctexttt%7BLuego%3A%20%7D%5C%5C%20%5C%5C%0A%5Ccsc%5Cleft%5Bx%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%284k%2B1%29%5Cright%5D%3D%5Csec%20x%5C%5C%20%5C%5C%0A%5Ctexttt%7Bcon%20la%20condici%5C%27on%3A%20%7D%5C%5C%20%5C%5C%0Ax%2B%5Cdfrac%7B%5Cpi%7D%7B2%7D%284k%2B1%29%5Cneq%20n%5Cpi%20%5Cwedge%20x%5Cneq%20%5Cdfrac%7B%5Cpi%7D%7B2%7D%282m%2B1%29%5C%5C%20%5C%5C%0Ax%5Cneq%20n%5Cpi%20-%5Cdfrac%7B%5Cpi%7D%7B2%7D%284k%2B1%29%5Cwedge%20x%5Cneq%20%5Cdfrac%7B%5Cpi%7D%7B2%7D%282m%2B1%29%5C%5C%20%5C%5C%0Ax%5Cneq%20%5Cdfrac%7B%5Cpi%7D%7B2%7D%282n-4k-1%29%5Cwedge%20x%5Cneq%20%5Cdfrac%7B%5Cpi%7D%7B2%7D%282m%2B1%29%5C%5C%20%5C%5C%20%5C%5C%0A%5Cboxed%7Bx%5Cneq%20%5Cdfrac%7B%5Cpi%7D%7B2%7D%282m%2B1%29%7D" />
Ahora resolvamos la ecuación
<img src="https://tex.z-dn.net/?f=%28x%2B40%5C%C2%B0%29-%285x%2B20%5C%C2%B0%29%3D90%5C%C2%B0%284k%2B1%29~%2C~%20k%5Cin%20%5Cmathbb%7BZ%7D%5C%5C%20%5C%5C%20%0A-4x%2B20%5C%C2%B0%3D360%5C%C2%B0k%2B90%5C%C2%B0%5C%5C%20%5C%5C%20%0A-4x%3D360%5C%C2%B0k%2B70%5C%C2%B0%5C%5C%20%5C%5C%20%0Ax%3D-90%5C%C2%B0k-17.5%5C%C2%B0%5C%5C%20%5C%5C%0Ax%3D90%5C%C2%B0n-17.5%5C%C2%B0~%2C~n%5Cin%20%5Cmathbb%7BZ%7D%5C%5C%20%5C%5C%0A%5Ctexttt%7BPor%20ejemplo%20en%20el%20primer%20cuadrante%20tenemos%20a%20%7Dx%20%3D%2090%5C%C2%B0-17.5%5C%C2%B0%5C%5C%20%5C%5C%0Ax%3D72.5%5C%C2%B0%3D72%5C%C2%B030%27" />.
Solo se igualan los que están en paréntesis :
5x + 20 = x + 40
4x = 20
x = 5.
Demostrar. Tanx + cotx = secx. Coscx (tanx = senx / cosx y cotx = cosx / senx) senx / cosx + cosx / senx = secx. Cscx (sen²x + cos²x) / senx cosx = secx . Cscx pero sen²x + cos²x = 1 Por identidad fundamental 1 /…
Respuesta : Para simplificar esta expresión debemos aplicar identidades trigonométricas, tal que : Sec(x)·(Csc(x) - 1) + Cos(x)·[(Sec(2x) - Csc(x)] Llevamos todo a Sen(x) y Cos(x), sabiendo que Sec(x) = 1 / Cos(x) y Csc…