第13章 13.18

(1)

$\sqrt{3}\cos x - \sin x = 1 \qquad 2\left(\frac{\sqrt{3}}{2}\cos x - \frac{1}{2}\sin x\right) = 1 \qquad 2\sin\left(\frac{\pi}{3} - x\right) = 1 \qquad \sin\left(\frac{\pi}{3} - x\right) = \frac{1}{2}$

これから

$x = \frac{\pi}{6} + 2n\pi, -\frac{\pi}{2} + 2n\pi$

(2)

$\sin 2x = \sin x \qquad 2\sin x \cdot \cos x = \sin x \qquad \sin x(2\cos x - 1) = 0 \qquad \sin x = 0 \qquad \cos x = \frac{1}{2}$

これから

$x = n\pi, \pm\frac{\pi}{3} + 2n\pi$

(3)

$\cos 2x = \sin x + \cos x \qquad \cos^2 x - \sin^2 x = (\cos x - \sin x)(\cos x + \sin x) = \sin x + \cos x$

$\cos x - \sin x = 1 \qquad \cos x + \sin x = 0$

これから

$x = -\frac{\pi}{4} + n\pi, 2n\pi, -\frac{\pi}{2} + 2n\pi$

(4)

$\cos x + \cos 2x + \cos 3x = 0 \qquad 2\cos 2x \cdot \cos x + \cos 2x = 0 \qquad \cos 2x(2\cos x + 1) = 0$

$\cos 2x = 0 \qquad \cos x = -\frac{1}{2}$

これから

$x = \pm\frac{\pi}{4} + 2n\pi, \pm\frac{3}{4}\pi + 2n\pi, \pm\frac{2}{3}\pi + 2n\pi$