Esercizio 18 ripasso goniometria e trigonometria

Ripasso goniometria e trigonometria

Home » Esercizio 18 ripasso goniometria e trigonometria
Generic selectors
Exact matches only
Search in title
Search in content
Post Type Selectors
post
page

 

Esercizio 18   (\bigstar\largewhitestar\largewhitestar\largewhitestar\largewhitestar). Semplificare le seguenti espressioni

(1)   \begin{equation*} \frac{\displaystyle\cos(-\pi)+\cos\left(-\frac{3\pi}{2}\right)-a}{(a+b)\sin\pi+a}+ \frac{\displaystyle\sin\left(-\frac{\pi}{2}\right)+b\cos\pi}{\displaystyle a-a\sin\frac{\pi}{2}+b\cos 0}+2, \end{equation*}

(2)   \begin{equation*} \frac{\displaystyle a\left(a\sin\frac{\pi}{2}+b\cos\frac{5\pi}{2}\right)-b\left(a\cos\frac{9\pi}{2}+b\sin\frac{13\pi}{2}\right)} {a(\sin 0+\cos 4\pi)-b(\sin\pi+\cos 5\pi)}, \end{equation*}

dove a e b sono numeri reali.

 

Svolgimento. Semplifichiamo (1) ottenendo

    \[\begin{aligned} &\frac{\displaystyle\cos(-\pi)+\cos\left(-\frac{3\pi}{2}\right)-a}{(a+b)\sin\pi+a}+ \frac{\displaystyle\sin\left(-\frac{\pi}{2}\right)+b\cos\pi}{\displaystyle a-a\sin\frac{\pi}{2}+b\cos 0}+2=\\ &=\frac{-1+0-a}{(a+b)0+a}+\frac{-1+b(-1)}{a-a(1)+b(1)}+2=\\ &=\frac{-1-a}{a}+\frac{-1-b}{b}+2=\\ &=-\frac{a+b}{ab}. \end{aligned}\]

Notiamo che (1) è definita per a,b\in\mathbb{R}\setminus\{0\}.

Semplifichiamo (2) ottenendo

    \[\begin{aligned} &\frac{\displaystyle\cos(-\pi)+\cos\left(-\frac{3\pi}{2}\right)-a}{(a+b)\sin\pi+a}+ \frac{\displaystyle\sin\left(-\frac{\pi}{2}\right)+b\cos\pi}{\displaystyle a-a\sin\frac{\pi}{2}+b\cos 0}+2=\\ &=\frac{-1+0-a}{(a+b)0+a}+\frac{-1+b(-1)}{a-a(1)+b(1)}+2=\\ &=\frac{-1-a}{a}+\frac{-1-b}{b}+2=-\frac{a+b}{ab}. \end{aligned}\]

    \[\frac{\displaystyle\cos(-\pi)+\cos\left(-\frac{3\pi}{2}\right)-a}{(a+b)\sin\pi+a}+ \frac{\displaystyle\sin\left(-\frac{\pi}{2}\right)+b\cos\pi}{\displaystyle a-a\sin\frac{\pi}{2}+b\cos 0}+2\]

    \[=\frac{-1+0-a}{(a+b)0+a}+\frac{-1+b(-1)}{a-a(1)+b(1)}+2=\frac{-1-a}{a}+\frac{-1-b}{b}+2=-\frac{a+b}{ab}.\]

Notiamo che (2) è definita per a\neq -b.

 

Fonte: Ignota