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Sagot :
Para resolver la pregunta dada, comenzamos conociendo que [tex]\( \sin \theta = \frac{60}{61} \)[/tex].
1. Cálculo de [tex]\( \cos \theta \)[/tex]:
Usamos la identidad pitagórica:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]
Sabemos que [tex]\( \sin \theta = \frac{60}{61} \)[/tex], entonces:
[tex]\[ \sin^2(\theta) = \left( \frac{60}{61} \right)^2 = \frac{3600}{3721} \][/tex]
Sustituimos en la identidad pitagórica:
[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{3600}{3721} = \frac{3721}{3721} - \frac{3600}{3721} = \frac{121}{3721} \][/tex]
Entonces:
[tex]\[ \cos(\theta) = \sqrt{\frac{121}{3721}} = \frac{\sqrt{121}}{\sqrt{3721}} = \frac{11}{61} \][/tex]
2. Cálculo de [tex]\( \sec \theta \)[/tex]:
La función secante es el inverso de la función coseno:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{11}{61}} = \frac{61}{11} \approx 5.545454545454545 \][/tex]
3. Cálculo de [tex]\( \tan \theta \)[/tex]:
La función tangente es el cociente entre la función seno y la función coseno:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{60}{61}}{\frac{11}{61}} = \frac{60}{11} \approx 5.454545454545454 \][/tex]
4. Cálculo de [tex]\( P \)[/tex]:
Sumamos los valores de la secante y la tangente:
[tex]\[ P = \sec \theta + \tan \theta = 5.545454545454545 + 5.454545454545454 \approx 11 \][/tex]
Por lo tanto, el valor de [tex]\( P \)[/tex] es [tex]\( 11 \)[/tex].
La respuesta correcta es: a) [tex]\( 11 \)[/tex]
1. Cálculo de [tex]\( \cos \theta \)[/tex]:
Usamos la identidad pitagórica:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]
Sabemos que [tex]\( \sin \theta = \frac{60}{61} \)[/tex], entonces:
[tex]\[ \sin^2(\theta) = \left( \frac{60}{61} \right)^2 = \frac{3600}{3721} \][/tex]
Sustituimos en la identidad pitagórica:
[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{3600}{3721} = \frac{3721}{3721} - \frac{3600}{3721} = \frac{121}{3721} \][/tex]
Entonces:
[tex]\[ \cos(\theta) = \sqrt{\frac{121}{3721}} = \frac{\sqrt{121}}{\sqrt{3721}} = \frac{11}{61} \][/tex]
2. Cálculo de [tex]\( \sec \theta \)[/tex]:
La función secante es el inverso de la función coseno:
[tex]\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{11}{61}} = \frac{61}{11} \approx 5.545454545454545 \][/tex]
3. Cálculo de [tex]\( \tan \theta \)[/tex]:
La función tangente es el cociente entre la función seno y la función coseno:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{60}{61}}{\frac{11}{61}} = \frac{60}{11} \approx 5.454545454545454 \][/tex]
4. Cálculo de [tex]\( P \)[/tex]:
Sumamos los valores de la secante y la tangente:
[tex]\[ P = \sec \theta + \tan \theta = 5.545454545454545 + 5.454545454545454 \approx 11 \][/tex]
Por lo tanto, el valor de [tex]\( P \)[/tex] es [tex]\( 11 \)[/tex].
La respuesta correcta es: a) [tex]\( 11 \)[/tex]
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