Kpl.1.3

Esim. Laske ilmna laskinta

$\sin\ \left(-240°\right)=-\sin240°=-\left(-\frac{\sqrt[]{3}}{2}\right)=\frac{\sqrt[]{3}}{2}$

b)
$\cos\left(-\frac{7\pi}{6}\right)=\cos\left(\frac{7\pi}{6}\right)=-\frac{\sqrt[]{3}}{2}$

c) Osoita, että $\sin\frac{\pi}{5}=\sin\frac{4\pi}{5}$

$\sin\frac{\pi}{5}=\sin\left(\pi-\frac{\pi}{5}\right)=\sin\left(\frac{5\pi}{5}-\frac{\pi}{5}\right)=\sin\frac{4\pi}{5}$

Lause: Kaksinkertaisen kulman sini ja kosini

$\sin2\alpha=2\sin\alpha\cos\alpha$

$\cos2\alpha=\cos^2\alpha-\sin^2\alpha\ \ \ \ \ \left(\sin^2\alpha+\cos^2\alpha=1\right)$
$\cos2\alpha=1-2\sin^2\alpha$
$\cos2\alpha=2\cos^2\alpha-1$

Esim. Tidetään, että $\cos\alpha=\frac{4}{5}$ . Määritä cos2α ja sin2α

$\cos2\alpha=2\cdot\left(\frac{4}{5}\right)^2-1=2\cdot\frac{16}{25}=\frac{32}{25}-1=1\frac{7}{25}-1=\frac{7}{25}=0{,}28$

$\left(\sin\alpha\right)^2+\left(\cos\alpha\right)^2=1$
$\left(\sin\alpha\right)^2=1-\left(\cos\alpha\right)^2$
$\left(\sin\alpha\right)^2=1-\left(\frac{4}{5}\right)^2$
$\left(\sin\alpha\right)^2=1-\frac{16}{25}$
$\left(\sin\alpha\right)^2=\frac{9}{25}\ \ \ \ \ \left|\right|\sqrt[]{}$
$\sin\alpha=\pm\frac{3}{5}=\pm0{,}6$

$\sin2\alpha=2\sin\alpha\cos\alpha$

$\sin2\alpha=2\cdot\frac{3}{5}\cdot\frac{4}{5}=0{,}96$

tai

$\sin2\alpha=2\cdot\left(-\frac{3}{5}\right)\cdot\frac{4}{5}=-0{,}96$