Lecture notes
FORMULARIO : FUNCIONES ELEMENTALES
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FORMULARIO DE : - FUNCIONES ELEMENTALES
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NÚMEROS COMPLEJOS: FORMULARIO▪ IDENTIDADES PITAGÓRICAS:
𝑠𝑒𝑛2 𝑥 + 𝑐𝑜𝑠 2 𝑥 = 1, 𝑡𝑎𝑛2 𝑥 + 1 = 𝑠𝑒𝑐 2 𝑥, 𝑐𝑜𝑡 2 𝑥 + 1 = 𝑐𝑜𝑠𝑒𝑐 2 𝑥
▪ SUMA Y DIFERENCIA DE ÁNGULOS:
𝑠𝑒𝑛(𝑥 ± 𝑦) = 𝑠𝑒𝑛 𝑥 𝑐𝑜𝑠𝑦 ± 𝑐𝑜𝑠 𝑥 𝑠𝑒𝑛𝑦
𝑐𝑜𝑠(𝑥 ± 𝑦) = 𝑐𝑜𝑠 𝑥 𝑐𝑜𝑠𝑦 ∓ 𝑠𝑒𝑛 𝑥 𝑠𝑒𝑛𝑦
𝑡𝑎𝑛 𝑥 ± 𝑡𝑎𝑛 𝑦
𝑡𝑎𝑛(𝑥 ± 𝑦) =
1 ∓ 𝑡𝑎𝑛 𝑥 𝑡𝑎𝑛 𝑦
▪ ÁNGULO DOBLE:
2 𝑡𝑎𝑛 𝑥
𝑠𝑒𝑛 2𝑥 = 2 𝑠𝑒𝑛 𝑥 𝑐𝑜𝑠 𝑥, 𝑐𝑜𝑠 2𝑥 = 𝑐𝑜𝑠 2 𝑥 − 𝑠𝑒𝑛2 𝑥, tan 2𝑥 =
1 − 𝑡𝑎𝑛2 𝑥
▪ ÁNGULO MITAD:
𝑥 1 − 𝑐𝑜𝑠 2𝑥 𝑥 1 + 𝑐𝑜𝑠 2𝑥 𝑥 1 − 𝑐𝑜𝑠 2𝑥
𝑠𝑒𝑛 =ඨ , 𝑐𝑜𝑠 =ඨ , 𝑡𝑎𝑛 =ඨ
2 2 2 2 2 1 + 𝑐𝑜𝑠 2𝑥
▪ EN FUNCIÓN DE LA TANGENTE DEL ÁNGULO MITAD:
𝑥 𝑥 𝑥
2 𝑡𝑎𝑛 2 1 − 𝑡𝑎𝑛2 2 2 𝑡𝑎𝑛 2
𝑠𝑒𝑛 𝑥 = 𝑥 , 𝑐𝑜𝑠 𝑥 = 𝑥 , 𝑡𝑎𝑛 𝑥 = 𝑥
1 + 𝑡𝑎𝑛2 1 + 𝑡𝑎𝑛2 1 − 𝑡𝑎𝑛2
2 2 2
▪ TRANSFORMACIÓN DE SUMAS EN PRODUCTOS:
𝑥 ± 𝑦 𝑥 ∓ 𝑦
𝑠𝑒𝑛 𝑥 ± 𝑠𝑒𝑛 𝑦 = 2 𝑠𝑒𝑛 cos
2 2
𝑥+𝑦 𝑥−𝑦
cos 𝑥 + cos 𝑦 = 2 cos 𝑐𝑜𝑠
2 2
𝑥+𝑦 𝑥−𝑦
𝑐𝑜𝑠 𝑥 − cos 𝑦 = −2 𝑠𝑒𝑛 𝑠𝑒𝑛
2 2
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