Nichols Gardner
05/01/2024 · Primary School
(4) \( \operatorname{sen}\left(30^{\circ}+p\right)-\cos \left(60^{\circ}+\beta\right)=\sqrt{3} \operatorname{sen} \beta \) (5) \( \operatorname{sen}^{3} x+\cos ^{3} x=(1-\operatorname{sen} x \cos x)(\operatorname{sen} x+\cos x) \) Demostración
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para (4) y (5)
### Demostración para (4)
Para demostrar la igualdad \( \operatorname{sen}\left(30^{\circ}+p\right)-\cos \left(60^{\circ}+\beta\right)=\sqrt{3} \operatorname{sen} \beta \), utilizaremos las propiedades de los trigonométricos y la identidad de seno y coseno sumas.
1. **Expresión de \(\operatorname{sen}(30^{\circ}+p)\):**
\[
\operatorname{sen}(30^{\circ}+p) = \operatorname{sen}30^{\circ}\cos p + \cos30^{\circ}\sin p
\]
\[
= \frac{1}{2}\cos p + \frac{\sqrt{3}}{2}\sin p
\]
2. **Expresión de \(\cos(60^{\circ}+\beta)\):**
\[
\cos(60^{\circ}+\beta) = \cos60^{\circ}\cos\beta - \sin60^{\circ}\sin\beta
\]
\[
= \frac{1}{2}\cos\beta - \frac{\sqrt{3}}{2}\sin\beta
\]
3. **Sustitución en la igualdad:**
\[
\operatorname{sen}(30^{\circ}+p) - \cos(60^{\circ}+\beta) = \left(\frac{1}{2}\cos p + \frac{\sqrt{3}}{2}\sin p\right) - \left(\frac{1}{2}\cos\beta - \frac{\sqrt{3}}{2}\sin\beta\right)
\]
\[
= \frac{1}{2}\cos p + \frac{\sqrt{3}}{2}\sin p - \frac{1}{2}\cos\beta + \frac{\sqrt{3}}{2}\sin\beta
\]
4. **Simplificación:**
\[
= \frac{1}{2}(\cos p - \cos\beta) + \frac{\sqrt{3}}{2}(\sin p + \sin\beta)
\]
5. **Uso de la identidad de seno:**
\[
\cos p - \cos\beta = -2\sin\left(\frac{p+\beta}{2}\right)\sin\left(\frac{p-\beta}{2}\right)
\]
\[
\sin p + \sin\beta = 2\sin\left(\frac{p+\beta}{2}\right)\cos\left(\frac{p-\beta}{2}\right)
\]
6. **Sustitución en la expresión simplificada:**
\[
= \frac{1}{2}\left(-2\sin\left(\frac{p+\beta}{2}\right)\sin\left(\frac{p-\beta}{2}\right)\right) + \frac{\sqrt{3}}{2}\left(2\sin\left(\frac{p+\beta}{2}\right)\cos\left(\frac{p-\beta}{2}\right)\right)
\]
\[
= -\sin\left(\frac{p+\beta}{2}\right)\sin\left(\frac{p-\beta}{2}\right) + \sqrt{3}\sin\left(\frac{p+\beta}{2}\right)\cos\left(\frac{p-\beta}{2}\right)
\]
7. **Uso de la identidad de seno:**
\[
\sin\left(\frac{p+\beta}{2}\right)\cos\left(\frac{p-\beta}{2}\right) = \frac{1}{2}\sin(p+\beta)
\]
\[
\sin\left(\frac{p+\beta}{2}\right)\sin\left(\frac{p-\beta}{2}\right) = \frac{1}{2}\left(\cos(p-\beta) - \cos(p+\beta)\right)
\]
8. **Sustitución en la expresión simplificada:**
\[
= -\frac{1}{2}\left(\cos(p-\beta) - \cos(p+\beta)\right) + \sqrt{3}\cdot\frac{1}{2}\sin(p+\beta)
\]
\[
= -\frac{1}{2}\cos(p-\beta) + \frac{1}{2}\cos(p+\beta) + \frac{\sqrt{3}}{2}\sin(p
Quick Answer
Demostración para (4) y (5) completada.
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