1. Probleem: Leia arvu $\cos 2\alpha$ täpne väärtus, kui $\sin \alpha = \frac{3}{5}$ ja $\frac{\pi}{2} < \alpha < \pi$. 2. Kasutame valemit: $$\cos 2\alpha = 1 - 2\sin^2 \alpha$$ või $$\cos 2\alpha = 2\cos^2 \alpha - 1$$. 3. Kuna $\sin \alpha = \frac{3}{5}$, arvutame $\cos \alpha$ kasutades Pythagorase teoreemi: $$\cos \alpha = \pm \sqrt{1 - \sin^2 \alpha} = \pm \sqrt{1 - \left(\frac{3}{5}\right)^2} = \pm \sqrt{1 - \frac{9}{25}} = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}$$. 4. Kuna $\frac{\pi}{2} < \alpha < \pi$, on $\alpha$ teises kvadrandis, kus $\cos \alpha$ on negatiivne, seega: $$\cos \alpha = -\frac{4}{5}$$. 5. Arvutame $\cos 2\alpha$ valemiga $$\cos 2\alpha = 2\cos^2 \alpha - 1$$: $$\cos 2\alpha = 2 \left(-\frac{4}{5}\right)^2 - 1 = 2 \cdot \frac{16}{25} - 1 = \frac{32}{25} - 1 = \frac{32}{25} - \frac{25}{25} = \frac{7}{25}$$. 6. Vastus: $$\cos 2\alpha = \frac{7}{25}$$.