高専の数学解説

第13章 13.13

sin⁡x=45cos⁡x=−35(π2<x<π)\sin x = \frac{4}{5} \qquad \cos x = -\frac{3}{5} \qquad (\frac{\pi}{2} < x < \pi)

(1)

sin⁡2x2=12(1−cos⁡x)=12(1+35)=12⋅85=45\sin^2\frac{x}{2} = \frac{1}{2}(1-\cos x) = \frac{1}{2}\left(1+\frac{3}{5}\right) = \frac{1}{2}\cdot\frac{8}{5} = \frac{4}{5}

sin⁡x2=25\sin\frac{x}{2} = \frac{2}{\sqrt{5}}

(2)

cos⁡2x2=12(1+cos⁡x)=12(1−35)=12⋅25=15\cos^2\frac{x}{2} = \frac{1}{2}(1+\cos x) = \frac{1}{2}\left(1-\frac{3}{5}\right) = \frac{1}{2}\cdot\frac{2}{5} = \frac{1}{5}

cos⁡x2=15\cos\frac{x}{2} = \frac{1}{\sqrt{5}}

(3)

tan⁡x2=sin⁡x2cos⁡x2=2\tan\frac{x}{2} = \frac{\sin\frac{x}{2}}{\cos\frac{x}{2}} = 2

解説: r31bn1z