RELACIONES ENTRE FUNCIONES HIPERBÓLICAS 2 2 • ch x − sh x = 1 • th x = • coth x = 2 2 • 1 − th x = sec h x sh x ch x • cos ech x = 1 th x 2 2 • coth x − 1 = cos ec x 1 sh x FUNCIONES DE ÁNGULOS NEGATIVOS • sh (− x ) = − sh x • ch (− x ) = ch x • th (− x ) = − th x FUNCIONES DE LA SUMA Y DIFERENCIA • sh (a ± b) = sh a ch b ± ch a sh b • ch (a ± b) = cha ch b ± sh a sh b • th(a ± b) = th a ± th b 1 ± th a th b FUNCIONES DEL ÁNGULO DOBLE 2 • sh 2a = 2 sh a ch a 2 2 2 • ch 2a = ch a + sh a = 1 + 2 sh a = 2 ch a − 1 SUMA, DIFERENCIA Y PRODUCTO DE LAS FUNCIONES HIPERBÓLICAS a+b a−b ⋅ ch 2 2 a+b a−b • ch a + ch b = 2 ⋅ ch ⋅ ch 2 2 1 • sh a ⋅ sh b = [ch (a + b) − ch (a − b)] 2 1 • sh a ⋅ ch b = [sh (a + b) + sh (a − b)] 2 • sh a + sh b = 2 ⋅ sh a+b a−b ⋅ sh 2 2 a+b a−b • ch a − ch b = 2 ⋅ sh ⋅ sh 2 2 1 • ch a ⋅ ch b = [ch (a + b) + ch (a − b)] 2 • sh a − sh b = 2 ⋅ ch EXPRESIONES DE EULER • sh a = e a −e 2 −a • ch a = e a +e 2 −a • th a = e e −a a −e a +e −a