Peterson Mann
04/12/2024 · Senior High School
ngulos, identidades para ángulos dob RECUERDA > Enuncia las identidades que co- rresponden con las siguientes expresiones. a. \( \operatorname{sen}(\alpha+\beta)= \) b. \( \cos (\alpha-\beta)= \) c. \( \tan (\alpha+\beta)= \) d. \( \operatorname{sen}(2 \beta)= \) e. \( \cos (2 \beta)= \)
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Aquí tienes las identidades trigonométricas correspondientes a las expresiones que mencionas:
a. \( \operatorname{sen}(\alpha+\beta) = \operatorname{sen}(\alpha) \cdot \operatorname{cos}(\beta) + \operatorname{cos}(\alpha) \cdot \operatorname{sen}(\beta) \)
b. \( \cos(\alpha-\beta) = \operatorname{cos}(\alpha) \cdot \operatorname{cos}(\beta) + \operatorname{sen}(\alpha) \cdot \operatorname{sen}(\beta) \)
c. \( \tan(\alpha+\beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \cdot \tan(\beta)} \)
d. \( \operatorname{sen}(2\beta) = 2 \cdot \operatorname{sen}(\beta) \cdot \operatorname{cos}(\beta) \)
e. \( \cos(2\beta) = \operatorname{cos}^2(\beta) - \operatorname{sen}^2(\beta) \)
También se puede expresar como:
\( \cos(2\beta) = 2 \cdot \operatorname{cos}^2(\beta) - 1 \)
o
\( \cos(2\beta) = 1 - 2 \cdot \operatorname{sen}^2(\beta) \)
Estas identidades son fundamentales en trigonometría y se utilizan frecuentemente en diversas aplicaciones matemáticas.
Quick Answer
a. \( \operatorname{sen}(\alpha+\beta) = \operatorname{sen}(\alpha) \cdot \operatorname{cos}(\beta) + \operatorname{cos}(\alpha) \cdot \operatorname{sen}(\beta) \)
b. \( \cos(\alpha-\beta) = \operatorname{cos}(\alpha) \cdot \operatorname{cos}(\beta) + \operatorname{sen}(\alpha) \cdot \operatorname{sen}(\beta) \)
c. \( \tan(\alpha+\beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha) \cdot \tan(\beta)} \)
d. \( \operatorname{sen}(2\beta) = 2 \cdot \operatorname{sen}(\beta) \cdot \operatorname{cos}(\beta) \)
e. \( \cos(2\beta) = \operatorname{cos}^2(\beta) - \operatorname{sen}^2(\beta) \) o \( 2 \cdot \operatorname{cos}^2(\beta) - 1 \) o \( 1 - 2 \cdot \operatorname{sen}^2(\beta) \)
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