Find the best solutions to your problems with the help of IDNLearn.com's expert users. Get comprehensive answers to all your questions from our network of experienced experts.
Sagot :
Para resolver las expresiones dadas y encontrar el valor numérico de [tex]\( x \)[/tex], utilizamos las identidades trigonométricas y los valores de las funciones trigonométricas para los ángulos dados. Vamos a resolver cada parte por separado.
### Parte a:
La expresión dada es:
[tex]\[ x = \frac{-3 \cdot \sin 315^\circ \cdot \tan 135^\circ}{\sin 510^\circ - 3 \cdot \sin 1290^\circ} \][/tex]
1. Calculamos [tex]\(\sin 315^\circ\)[/tex]:
[tex]\(\sin 315^\circ = \sin(360^\circ - 45^\circ) = -\sin 45^\circ = -\frac{\sqrt{2}}{2}\)[/tex]
2. Calculamos [tex]\(\tan 135^\circ\)[/tex]:
[tex]\(\tan 135^\circ = \tan(180^\circ - 45^\circ) = -\tan 45^\circ = -1\)[/tex]
3. Calculamos [tex]\(\sin 510^\circ\)[/tex]:
[tex]\(\sin 510^\circ = \sin(510^\circ - 360^\circ) = \sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2}\)[/tex]
4. Calculamos [tex]\(\sin 1290^\circ\)[/tex]:
[tex]\(\sin 1290^\circ = \sin(1290^\circ - 3 \times 360^\circ) = \sin 210^\circ = \sin(180^\circ + 30^\circ) = -\sin 30^\circ = -\frac{1}{2}\)[/tex]
Ahora sustituyendo estos valores en la expresión original:
[tex]\[ x = \frac{-3 \cdot \left(-\frac{\sqrt{2}}{2}\right) \cdot (-1)}{\frac{1}{2} - 3 \cdot \left(-\frac{1}{2}\right)} \][/tex]
Simplificando el numerador y el denominador:
[tex]\[ x = \frac{-3 \cdot \frac{\sqrt{2}}{2} \cdot (-1)}{\frac{1}{2} + \frac{3}{2}} \][/tex]
[tex]\[ x = \frac{-3 \cdot \frac{\sqrt{2}}{2}}{2} \][/tex]
[tex]\[ x = \frac{-3 \sqrt{2}}{4} \][/tex]
Finalmente, el resultado numérico es:
[tex]\[ x \approx -1.0606601717798239 \][/tex]
### Parte b:
La expresión dada es:
[tex]\[ x = \frac{2 \cdot \sin 120^\circ}{\cot 30^\circ - 4 \cdot \sin 45^\circ} \][/tex]
1. Calculamos [tex]\(\sin 120^\circ\)[/tex]:
[tex]\(\sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex]
2. Calculamos [tex]\(\cot 30^\circ\)[/tex]:
[tex]\(\cot 30^\circ = \frac{1}{\tan 30^\circ} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}\)[/tex]
3. Calculamos [tex]\(\sin 45^\circ\)[/tex]:
[tex]\(\sin 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]
Ahora sustituyendo estos valores en la expresión original:
[tex]\[ x = \frac{2 \cdot \frac{\sqrt{3}}{2}}{\sqrt{3} - 4 \cdot \frac{\sqrt{2}}{2}} \][/tex]
Simplificando el numerador y el denominador:
[tex]\[ x = \frac{\sqrt{3}}{\sqrt{3} - 2\sqrt{2}} \][/tex]
Ahora racionalizando el denominador:
[tex]\[ x = \frac{\sqrt{3} (\sqrt{3} + 2\sqrt{2})}{(\sqrt{3} - 2\sqrt{2})(\sqrt{3} + 2\sqrt{2})} \][/tex]
[tex]\[ x = \frac{3 + 2\sqrt{6}}{3 - 8} \][/tex]
[tex]\[ x = \frac{3 + 2\sqrt{6}}{-5} \][/tex]
Finalmente, el resultado numérico es:
[tex]\[ x \approx -1.579795897113272 \][/tex]
Así, los valores numéricos de [tex]\( x \)[/tex] en las expresiones dadas son aproximadamente [tex]\(-1.0606601717798239\)[/tex] para la parte a y [tex]\(-1.579795897113272\)[/tex] para la parte b.
### Parte a:
La expresión dada es:
[tex]\[ x = \frac{-3 \cdot \sin 315^\circ \cdot \tan 135^\circ}{\sin 510^\circ - 3 \cdot \sin 1290^\circ} \][/tex]
1. Calculamos [tex]\(\sin 315^\circ\)[/tex]:
[tex]\(\sin 315^\circ = \sin(360^\circ - 45^\circ) = -\sin 45^\circ = -\frac{\sqrt{2}}{2}\)[/tex]
2. Calculamos [tex]\(\tan 135^\circ\)[/tex]:
[tex]\(\tan 135^\circ = \tan(180^\circ - 45^\circ) = -\tan 45^\circ = -1\)[/tex]
3. Calculamos [tex]\(\sin 510^\circ\)[/tex]:
[tex]\(\sin 510^\circ = \sin(510^\circ - 360^\circ) = \sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2}\)[/tex]
4. Calculamos [tex]\(\sin 1290^\circ\)[/tex]:
[tex]\(\sin 1290^\circ = \sin(1290^\circ - 3 \times 360^\circ) = \sin 210^\circ = \sin(180^\circ + 30^\circ) = -\sin 30^\circ = -\frac{1}{2}\)[/tex]
Ahora sustituyendo estos valores en la expresión original:
[tex]\[ x = \frac{-3 \cdot \left(-\frac{\sqrt{2}}{2}\right) \cdot (-1)}{\frac{1}{2} - 3 \cdot \left(-\frac{1}{2}\right)} \][/tex]
Simplificando el numerador y el denominador:
[tex]\[ x = \frac{-3 \cdot \frac{\sqrt{2}}{2} \cdot (-1)}{\frac{1}{2} + \frac{3}{2}} \][/tex]
[tex]\[ x = \frac{-3 \cdot \frac{\sqrt{2}}{2}}{2} \][/tex]
[tex]\[ x = \frac{-3 \sqrt{2}}{4} \][/tex]
Finalmente, el resultado numérico es:
[tex]\[ x \approx -1.0606601717798239 \][/tex]
### Parte b:
La expresión dada es:
[tex]\[ x = \frac{2 \cdot \sin 120^\circ}{\cot 30^\circ - 4 \cdot \sin 45^\circ} \][/tex]
1. Calculamos [tex]\(\sin 120^\circ\)[/tex]:
[tex]\(\sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2}\)[/tex]
2. Calculamos [tex]\(\cot 30^\circ\)[/tex]:
[tex]\(\cot 30^\circ = \frac{1}{\tan 30^\circ} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}\)[/tex]
3. Calculamos [tex]\(\sin 45^\circ\)[/tex]:
[tex]\(\sin 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]
Ahora sustituyendo estos valores en la expresión original:
[tex]\[ x = \frac{2 \cdot \frac{\sqrt{3}}{2}}{\sqrt{3} - 4 \cdot \frac{\sqrt{2}}{2}} \][/tex]
Simplificando el numerador y el denominador:
[tex]\[ x = \frac{\sqrt{3}}{\sqrt{3} - 2\sqrt{2}} \][/tex]
Ahora racionalizando el denominador:
[tex]\[ x = \frac{\sqrt{3} (\sqrt{3} + 2\sqrt{2})}{(\sqrt{3} - 2\sqrt{2})(\sqrt{3} + 2\sqrt{2})} \][/tex]
[tex]\[ x = \frac{3 + 2\sqrt{6}}{3 - 8} \][/tex]
[tex]\[ x = \frac{3 + 2\sqrt{6}}{-5} \][/tex]
Finalmente, el resultado numérico es:
[tex]\[ x \approx -1.579795897113272 \][/tex]
Así, los valores numéricos de [tex]\( x \)[/tex] en las expresiones dadas son aproximadamente [tex]\(-1.0606601717798239\)[/tex] para la parte a y [tex]\(-1.579795897113272\)[/tex] para la parte b.
Your engagement is important to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.
