Espressioni goniometriche (1)

[piptrip]

Potreste aiutarmi con le seguenti espressioni?

1. cos(a + 135°) - cos(225° - a) + cos(- a)

2. 2cot(3π/2 - a)cos(π + a) + 3sin^2(π + a)/cos(-π/2 - a) - cos(-3π/2 + a)cos(-a)/sin(a - π/2)

3. sin(2a - π/6) + 2cos^2(π/3 + a)

Grazie.

[ce8c0e31f941cc14ed8def9a9c5c3a60a8400d85]

Avendo ben presenti gli archi associati:

1.

[math]
\small
\begin{aligned}
& \dots \cos(\alpha + 135°) - \cos(225° - \alpha) + \cos(-\alpha) \\
& = \cos((\alpha - 45°) + 180°) - \cos((45° - \alpha) + 180°) + \cos(-\alpha) \\
& = - \cos(\alpha - 45°) + \cos(45° - \alpha) + \cos(-\alpha) \\
& = - \cos(\alpha - 45°) + \cos(\alpha - 45°) + \cos\alpha \\
& = \cos\alpha \; .
\end{aligned} \\
[/math]

2.

[math]
\small
\begin{aligned}
& \dots 2\cot\left(\frac{3}{2}\pi - \alpha\right)\cos(\pi + \alpha) + \frac{3\sin^2(\pi + \alpha)}{\cos\left(-\frac{\pi}{2} - \alpha\right)} - \frac{\cos\left(-\frac{3}{2}\pi + \alpha\right)\cos(-\alpha)}{\sin\left(\alpha - \frac{\pi}{2}\right)} \\

& = 2\tan\alpha(-\cos\alpha) + \frac{3(-\sin\alpha)^2}{-\sin\alpha} - \frac{(-\sin\alpha)\cos\alpha}{-\cos\alpha} \\

& = - 2\sin\alpha - 3\sin\alpha - \sin\alpha \\

& = - 6\sin\alpha \; .
\end{aligned} \\
[/math]

Avendo ben presenti le formule di addizione/sottrazione:

3.

[math]
\small
\begin{aligned}
& \dots \sin\left(2\alpha - \frac{\pi}{6}\right) + 2\cos^2\left(\frac{\pi}{3} + \alpha\right) \\

& = \sin(2\alpha)\cos\left(\frac{\pi}{6}\right) - \cos(2\alpha)\sin\left(\frac{\pi}{6}\right) + 2\left[\cos\left(\frac{\pi}{3}\right)\cos\alpha - \sin\left(\frac{\pi}{3}\right)\sin\alpha\right]^2 \\

& = \frac{\sqrt{3}}{2}\sin(2\alpha) - \frac{1}{2}\cos(2\alpha) + 2\left(\frac{1}{2}\cos\alpha - \frac{\sqrt{3}}{2}\sin\alpha\right)^2 \\

& = \sqrt{3}\sin\alpha\cos\alpha - \frac{1}{2}\cos^2\alpha + \frac{1}{2}\sin^2\alpha + \frac{1}{2}\cos^2\alpha + \frac{3}{2}\sin^2\alpha - \sqrt{3}\sin\alpha\cos\alpha \\

& = 2\sin^2\alpha \; .
\end{aligned} \\
[/math]

Spero sia sufficientemente chiaro. ;)