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MATU • Trigonometria
MATU_TREC_025
Compendio de Trigonometría
Enunciado
Paso 1:
Hallar el valor de: $\tan(\alpha + \gamma)$, sabiendo que: $\tan \alpha = \frac{1}{3}; \tan \beta = \frac{1}{4}; \tan(\gamma - \beta) = \frac{1}{5}$
Hallar el valor de: $\tan(\alpha + \gamma)$, sabiendo que: $\tan \alpha = \frac{1}{3}; \tan \beta = \frac{1}{4}; \tan(\gamma - \beta) = \frac{1}{5}$
Solución Paso a Paso
1. Desarrollo paso a paso:
Notamos que $(\alpha + \gamma) = (\alpha + \beta) + (\gamma - \beta)$.
Primero calculamos $\tan(\alpha + \beta)$:
$$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} = \frac{1/3 + 1/4}{1 - (1/3)(1/4)} = \frac{7/12}{11/12} = \frac{7}{11}$$
Ahora calculamos $\tan((\alpha + \beta) + (\gamma - \beta))$:
$$\tan(\alpha + \gamma) = \frac{7/11 + 1/5}{1 - (7/11)(1/5)} = \frac{(35 + 11)/55}{(55 - 7)/55} = \frac{46}{48}$$
Simplificando:
$$\tan(\alpha + \gamma) = \frac{23}{24}$$
2. Resultado final:
$$\tan(\alpha + \gamma) = \frac{23}{24}$$
Notamos que $(\alpha + \gamma) = (\alpha + \beta) + (\gamma - \beta)$.
Primero calculamos $\tan(\alpha + \beta)$:
$$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} = \frac{1/3 + 1/4}{1 - (1/3)(1/4)} = \frac{7/12}{11/12} = \frac{7}{11}$$
Ahora calculamos $\tan((\alpha + \beta) + (\gamma - \beta))$:
$$\tan(\alpha + \gamma) = \frac{7/11 + 1/5}{1 - (7/11)(1/5)} = \frac{(35 + 11)/55}{(55 - 7)/55} = \frac{46}{48}$$
Simplificando:
$$\tan(\alpha + \gamma) = \frac{23}{24}$$
2. Resultado final:
$$\tan(\alpha + \gamma) = \frac{23}{24}$$