←Vrati se na zadatke1. ZadatakIzračunajte sin4π8+cos43π8.\sin^4\frac\pi8+\cos^4\frac{3\pi}8.sin48π+cos483π.Prikaži rješenjeRješenjesin4π8+cos43π8=sin4π8+sin4(π2−3π8)=2sin4π8=2(sin2π8)2=2(1−cos(2⋅π8)2)2=12(1−22)2=34−22.\begin{aligned} \sin^4\frac\pi8+\cos^4\frac{3\pi}8 &=\sin^4\frac\pi8+\sin^4\left(\frac\pi2-\frac{3\pi}8\right)\\ &=2\sin^4\frac\pi8\\ &=2\left(\sin^2\frac\pi8\right)^2\\ &=2\left(\frac{1-\cos(2\cdot\frac\pi8)}{2}\right)^2\\ &=\frac12\left(1-\frac{\sqrt2}{2}\right)^2\\ &=\frac34-\frac{\sqrt2}{2}. \end{aligned}sin48π+cos483π=sin48π+sin4(2π−83π)=2sin48π=2(sin28π)2=2(21−cos(2⋅8π))2=21(1−22)2=43−22.Pomakni formulu lijevo ili desno.
