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TrigonometríaAutores: Sofía B. Llano Pérez y Thalía Almeida Romero
¿Qué es?
La trigonometría es una rama importante de las matemáticas dedicada al estudio de la relación entre
los lados y ángulos de un triángulo rectángulo y una circunferencia.
Razones trigonométricas
Sea α un ángulo agudo de vértice A en un triángulo rectángulo ABC ubicado en el centro de una circunferencia,
sean a y b los catetos y c la hipotenusa, se llama:
𝑎 𝑐𝑎𝑡. 𝑜𝑝𝑢𝑒𝑠𝑡𝑜
𝑠𝑒𝑛 𝛼 = =
𝑐 ℎ𝑖𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑎
𝑏 𝑐𝑎𝑡. 𝑎𝑑𝑦𝑎𝑐𝑒𝑛𝑡𝑒
𝑐𝑜𝑠 𝛼 = =
𝑐 ℎ𝑖𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑎
𝑎 𝑐𝑎𝑡. 𝑜𝑝𝑢𝑒𝑠𝑡𝑜
𝑡𝑎𝑛 𝛼 = =
𝑏 𝑐𝑎𝑡. 𝑎𝑑𝑦𝑎𝑐𝑒𝑛𝑡𝑒
𝑏 𝑐𝑎𝑡. 𝑎𝑑𝑦𝑎𝑐𝑒𝑛𝑡𝑒
𝑐𝑜𝑡 𝛼 = =
𝑎 𝑐𝑎𝑡. 𝑜𝑝𝑢𝑒𝑠𝑡𝑜
Nota. Las razones trigonométricas solo son aplicables a los ángulos agudos del triángulo rectángulo.
Identidades Ángulo duplo
1. 𝑠𝑒𝑛2 𝛼 + 𝑐𝑜𝑠 2 𝛼 = 1 𝑠𝑒𝑛 2𝛼 = 2 𝑠𝑒𝑛 𝛼 ∙ cos 𝛼
𝑠𝑒𝑛 𝛼 cos 2𝛼 = 𝑐𝑜𝑠 2 𝛼 − 𝑠𝑒𝑛2 𝛼
2. 𝑡𝑎𝑛 𝛼 = 𝑐𝑜𝑠 𝛼
2 tan 𝛼
𝑐𝑜𝑠 𝛼 tan 2𝛼 =
3. 𝑐𝑜𝑡 𝛼 = 𝑠𝑒𝑛 𝛼 1 − 𝑡𝑎𝑛2 𝛼
1
4. 1 + 𝑡𝑎𝑛2 𝛼 = 𝑐𝑜𝑠2 𝛼 Paridad
1
5. 1 + 𝑐𝑜𝑡 2 𝛼 = 𝑠𝑒𝑛2 𝛼 Para todo ángulo α se cumple:
1 𝑠𝑒𝑛 ሺ−αሻ = −𝑠𝑒𝑛 α
6. 𝑠𝑒𝑐 𝛼 = 𝑐𝑜𝑠 𝛼
𝑐𝑜𝑠 ሺ−αሻ = 𝑐𝑜𝑠 α
1
7. 𝑐𝑠𝑐 𝛼 = 𝑠𝑒𝑛 𝛼 𝑡𝑎𝑛 ሺ−αሻ = −𝑡𝑎𝑛 α
𝑐𝑜𝑡 ሺ−αሻ = −𝑐𝑜𝑡 α
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