(a) find the minimum value and the maximum value,
(b) find the period.
Solutions: (a) min. value: \(-1\), max. value: \(1\); (b) period: \(2\pi\)(a) find the minimum value and the maximum value,
(b) find the period.
Solutions: (a) min. value: \(-1\), max. value: \(3\); (b) period: \(2\pi\)(a) find the domain and range,
(b) find the period,
(c) write down minima and maxima.
Solutions: (a) domain: \(\mathbb{R}\), range: \([2,4]\); (b) period: \(\pi\); (c) min.: \((-\frac{\pi}{4}+k\pi,2)\), max.:\((\frac{\pi}{4}+k\pi,4),~~ k\in\mathbb{Z}\)(a) find \(x\)-intercepts,
(b) find the period,
(c) write down minima and maxima.
Solutions: (a) \(x\)-intercepts: \(\frac{k\pi}{3},~~ k\in\mathbb{Z}\); (b) period: \(\frac{2\pi}{3}\); (c) min.: \((-\frac{\pi}{6}+\frac{2k\pi}{3},-1)\), max.:\((\frac{\pi}{6}+\frac{2k\pi}{3},1),~~ k\in\mathbb{Z}\)(a) find all \(x\)-intercepts on \([0,2\pi]\),
(b) write down minima and maxima on \([0,2\pi]\).
Solutions: (a) \(x\)-intercepts: \(\frac{\pi}{6},~\frac{\pi}{2},~\frac{5\pi}{6},~\frac{7\pi}{6},~\frac{3\pi}{2},~\frac{11\pi}{6}\); (b) min.: \((\frac{\pi}{3},-1),~(\pi,-1),~(\frac{5\pi}{3},-1)\), max.:\((0,1),~(\frac{2\pi}{3},1),~(\frac{4\pi}{3},1),~(2\pi,1)\)(a) find \(y\)-intercept,
(b) find all \(x\)-intercepts on \([0,2\pi]\),
(c) write down the greatest and the least value, and state the smallest non-negative value of \(x\) for which they occur.
Solutions: (a) \(y\)-intercept: \(2\sqrt{3}\); (b) \(x\)-intercepts: \(\frac{2\pi}{3},~ \frac{5\pi}{3}\); (c) min. value \(-4\) occurs at \(x=\frac{7\pi}{6}\), max. value \(4\) occurs at \(x=\frac{\pi}{6}\)(a) find \(x\)-intercepts,
(b) find the vertical asymptotes.
Solutions: (a) \(x\)-intercepts: \(x=k\pi,~~ k\in\mathbb{Z}\); (b) vertical asymptotes: \(x=\frac{\pi}{2}+k\pi,~~ k\in\mathbb{Z}\)(a) find \(x\)-intercepts,
(b) find the vertical asymptotes.
Solutions: (a) \(x\)-intercepts: \(x=2k\pi,~~ k\in\mathbb{Z}\); (b) vertical asymptotes: \(x=\pi+2k\pi,~~ k\in\mathbb{Z}\)(a) \(y=|\sin x|\)
(b) \(y=|2\cos x+1|\)
(a) \(y=\sin |x|\)
(b) \(y=\tan |x|\)
(a) find \(x\)-intercepts,
(b) find the vertical asymptotes,
(c) find minima and maxima.
Solutions: (a) \(x\)-intercepts: \(x=k\pi,~~ k\in\mathbb{Z}\); (b) vertical asymptotes: \(x=\frac{3\pi}{2}+2k\pi,~~ k\in\mathbb{Z}\); (c) maxima: \((\frac{\pi}{2}+2k\pi,1),~~ k\in\mathbb{Z}\), minima don't exist(a) \(\sin120^\circ\)
(b) \(\cos135^\circ\)
(c) \(\tan150^\circ\)
Solutions: (a) \(\cdots=\frac{\sqrt{3}}{2}\); (b) \(\cdots=-\frac{\sqrt{2}}{2}\); (c) \(\cdots=-\frac{\sqrt{3}}{3}\)(a) \(\cos210^\circ\)
(b) \(\tan225^\circ\)
(c) \(\sin330^\circ\)
Solutions: (a) \(\cdots=-\frac{\sqrt{3}}{2}\); (b) \(\cdots=1\); (c) \(\cdots=-\frac{1}{2}\)(a) \(\sin405^\circ\)
(b) \(\tan480^\circ\)
(c) \(\cos540^\circ\)
(d) \(\sin1290^\circ\)
Solutions: (a) \(\cdots=\frac{\sqrt{2}}{2}\); (b) \(\cdots=-\sqrt{3}\); (c) \(\cdots=-1\); (d) \(\cdots=-\frac{1}{2}\)(a) \(\sin(-60^\circ)\)
(b) \(\tan(-135^\circ)\)
(c) \(\cos(-270^\circ)\)
Solutions: (a) \(\cdots=-\frac{\sqrt{3}}{2}\); (b) \(\cdots=1\); (c) \(\cdots=0\)(a) \(\sin\frac{7\pi}{6}\)
(b) \(\tan\frac{11\pi}{3}\)
(c) \(\cos(-7\pi)\)
Solutions: (a) \(\cdots=-\frac{1}{2}\); (b) \(\cdots=-\sqrt{3}\); (c) \(\cdots=-1\)(a) \(\cos190^\circ\)
(b) \(\sin500^\circ\)
(c) \(\tan\frac{7\pi}{5}\)
Solutions: (a) \(\cdots=-\cos10^\circ\); (b) \(\cdots=\sin40^\circ\); (c) \(\cdots=\tan\frac{2\pi}{5}=\tan72^\circ\)(a) \(\cos x\)
(b) \(\tan x\)
Solutions: (a) \(\cos x=-\frac{8}{17}\); (b) \(\tan x=-\frac{15}{8}\)(a) \(\sin x\)
(b) \(\tan x\)
Solutions: (a) \(\sin x=-\frac{\sqrt{5}}{3}\); (b) \(\tan x=\frac{\sqrt{5}}{2}\)(a) \(\cos x\)
(b) \(\sin x\)
Solutions: (a) \(\cos x=-\frac{1}{3}\); (b) \(\sin x=\frac{2\sqrt{2}}{3}\)(a) \({\displaystyle\frac{1-\cos^2 x}{\sin x}}\)
(b) \({\displaystyle\left(\frac{1}{\sin x}-\sin x\right)\cdot\frac{1}{\cos^2 x}}\)
(c) \({\displaystyle\left(\sin x-\frac{1}{\sin x}\right)\cdot\tan x}\)
Solutions: (a) \(\cdots=\sin x\); (b) \(\cdots=\frac{1}{\sin x}\); (c) \(\cdots=-\cos x\)(a) \({\displaystyle\frac{\tan^2 x}{\sin^2 x}\cdot\frac{1-\sin^2 x}{1+\tan^2 x}}\)
(b) \({\displaystyle\frac{{\displaystyle\frac{1}{\cos x}}-\cos x}{\tan x}+\frac{1}{\sin x(\tan^2 x+1)}}\)
(c) \({\displaystyle\left(\tan x+\frac{1}{\tan x}-\frac{1}{\sin x}\right)(1+\cos x)}\)
Solutions: (a) \(\cdots=\cos^2 x\); (b) \(\cdots=\frac{1}{\sin x}\); (c) \(\cdots=\tan x\)(a) \({\displaystyle\sin 2x\cdot\frac{{\displaystyle\frac{1}{\sin x}}-\sin x}{\cos x}}\)
(b) \({\displaystyle\frac{2\tan x-\sin 2x}{1-\cos 2x}}\)
(c) \({\displaystyle\frac{{\displaystyle\frac{1}{\cos 2x}}-\cos^2 x+\sin^2 x}{\tan 2x}}\)
(d) \({\displaystyle\frac{\tan x+\sin x}{\tan x \sin 2x}\cdot(1-\cos x)}\)
Solutions: (a) \(\cdots=2\cos^2 x\); (b) \(\cdots=\tan x\); (c) \(\cdots=\sin 2x\); (d) \(\cdots=\frac{1}{2}\tan x\)(a) \(\sin x=\frac{1}{2}\)
(b) \(\sin x=-\frac{\sqrt{2}}{2}\)
(c) \(\cos x=\frac{1}{2}\)
(d) \(\cos x=0\)
Solutions: (a) \(x_1=30^\circ,~ x_2=150^\circ\); (b) \(x_1=-45^\circ,~ x_2=-135^\circ\); (c) \(x_1=60^\circ,~ x_2=-60^\circ\); (d) \(x_1=90^\circ,~ x_2=-90^\circ\)(a) \(\sin x=\frac{1}{2}\)
(b) \(\sin x=\frac{\sqrt{2}}{2}\)
(c) \(\cos x=\frac{\sqrt{2}}{2}\)
(d) \(\cos x=-\frac{1}{2}\)
Solutions: (a) \(x_1=\frac{\pi}{6},~ x_2=\frac{5\pi}{6}\); (b) \(x_1=\frac{\pi}{4},~ x_2=\frac{3\pi}{4}\); (c) \(x_1=\frac{\pi}{4},~ x_2=\frac{7\pi}{4}\); (d) \(x_1=\frac{5\pi}{6},~ x_2=\frac{7\pi}{6}\)(a) \(\sin x=\frac{1}{2}\)
(b) \(\sin x=-1\)
(c) \(\cos x=\frac{\sqrt{3}}{2}\)
(d) \(\cos x=-\frac{1}{2}\)
Solutions: (a) \(x_1=\frac{\pi}{6}+2k\pi,~ x_2=\frac{5\pi}{6}+2k\pi,~~ (k\in\mathbb{Z})\); (b) \(x_1=-\frac{\pi}{2}+2k\pi,~~ (k\in\mathbb{Z})\); (c) \(x_1=\pm\frac{\pi}{6}+2k\pi,~~ (k\in\mathbb{Z})\); (d) \(x_1=\pm\frac{2\pi}{3}+2k\pi,~~ (k\in\mathbb{Z})\)(a) \(\sin 2x=\frac{1}{2}\)
(b) \(\sin \left(x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\)
(c) \(\cos 5x=-\frac{\sqrt{3}}{2}\)
(d) \(\cos \left(3x-\frac{\pi}{6}\right)=\sqrt{2}\)
Solutions: (a) \(x_1=\frac{\pi}{12}+k\pi,~ x_2=\frac{5\pi}{12}+k\pi,~~ (k\in\mathbb{Z})\); (b) \(x_1=2k\pi,~ x_2=\frac{\pi}{2}+2k\pi,~~ (k\in\mathbb{Z})\); (c) \(x_1=\pm\frac{\pi}{6}+\frac{2}{5}k\pi,~~ (k\in\mathbb{Z})\); (d) this equation has no solutions (cosine can't be greater than 1)(a) \(\sin 2x=\frac{1}{2}\)
(b) \(\sin \left(5x+40^\circ\right)=\frac{\sqrt{3}}{2}\)
(c) \(\cos 6x=-1\)
(d) \(\cos (x+25^\circ)=\frac{1}{3}\)
Solutions: (a) \(x_1=15^\circ+180^\circ k,~ x_2=75^\circ+180^\circ k,~~ (k\in\mathbb{Z})\); (b) \(x_1=4^\circ+72^\circ k,~ x_2=16^\circ+72^\circ k,~~ (k\in\mathbb{Z})\); (c) \(x_1=30^\circ+60^\circ k,~~ (k\in\mathbb{Z})\); (d) \(x_1\approx 45^\circ32'+360^\circ k,~ x_2\approx -95^\circ32'+360^\circ k,~~ (k\in\mathbb{Z})\)(a) \(\sin 2x=\frac{1}{2}\)
(b) \(\sin \left(3x-15^\circ30'\right)=0\)
(c) \(\sin \frac{3x}{2}=\frac{2}{3}\)
Solutions: (a) \(x_1=15^\circ,~ x_2=75^\circ,~ x_3=195^\circ,~ x_4=255^\circ\); (b) \(x_1=5^\circ10',~ x_2=65^\circ10',~ x_3=125^\circ10',~ x_4=185^\circ10',~ x_5=245^\circ10',~ x_6=305^\circ10'\); (c) \(x_1\approx27^\circ52',~ x_2\approx92^\circ8',~ x_3\approx267^\circ52',~ x_4\approx332^\circ8'\)(a) \(\tan x=1\)
(b) \(\tan x=\sqrt{3}\)
(c) \(\tan 2x=-1\)
(d) \(\tan 2x=3\)
Solutions: (a) \(x_1=\frac{\pi}{4}+k\pi,~~ (k\in\mathbb{Z})\); (b) \(x_1=\frac{\pi}{3}+k\pi,~~ (k\in\mathbb{Z})\); (c) \(x_1=-\frac{\pi}{8}+\frac{1}{2}k\pi,~~ (k\in\mathbb{Z})\); (d) \(x_1\approx0.6245+\frac{k\pi}{2},~~ (k\in\mathbb{Z})\)(a) \(\tan (x+25^\circ)=\frac{\sqrt{3}}{3}\)
(b) \(\tan 9x=0\)
(c) \(\tan (2x-72^\circ)=-\frac{1}{2}\)
Solutions: (a) \(x_1=5^\circ+180^\circ k,~~ (k\in\mathbb{Z})\); (b) \(x_1=20^\circ k,~~ (k\in\mathbb{Z})\); (c) \(x_1\approx22^\circ43'+90^\circ k,~~ (k\in\mathbb{Z})\)(a) \(\tan (x-15^\circ)=\sqrt{3}\)
(b) \(\tan \frac{3x}{2}=-3\)
Solutions: (a) \(x_1=75^\circ,~ x_2=-105^\circ\); (b) \(x_1\approx72^\circ17',~ x_2\approx-47^\circ43',~ x_3\approx-167^\circ43'\)(a) \(\cos^2 x-4\cos x+3=0\)
(b) \(2\sin^2 x-11\sin x=6\)
(c) \(2\sin^2 x+3\cos x=3\)
Solutions: (a) \(x=2k\pi~~ (k\in\mathbb{Z})\); (b) \(x_1=-\frac{\pi}{6}+2k\pi,~ x_2=-\frac{5\pi}{6}+2k\pi~~ (k\in\mathbb{Z})\); (c) \(x_1=2k\pi,~ x_2=\pm\frac{\pi}{3}+2k\pi~~ (k\in\mathbb{Z})\)(a) \(\sin^2 x+2\sin x=0\)
(b) \(\tan^2 x-\tan x=0\)
(c) \(2\sin x\cos x-2\sin x+\sqrt{3}\cos x-\sqrt{3}=0 \)
Solutions: (a) \(x=k\pi~~ (k\in\mathbb{Z})\); (b) \(x_1=k\pi,~ x_2=\frac{\pi}{4}+k\pi~~ (k\in\mathbb{Z})\); (c) \(x_1=2k\pi,~ x_2=-\frac{\pi}{3}+2k\pi,~ x_3=-\frac{2\pi}{3}+2k\pi~~ (k\in\mathbb{Z})\)(a) \(4\sin^2 x\cos x=8\cos x-7\sin2x\)
(b) \(\tan x\cos^2 x-\frac{5}{2}\tan x=-\frac{9}{2}\tan x\cos x\)
Solutions: (a) \(x_1=\frac{\pi}{2}+k\pi,~ x_2=\frac{\pi}{6}+2k\pi,~ x_2=\frac{5\pi}{6}+2k\pi~~ (k\in\mathbb{Z})\); (b) \(x_1=k\pi,~ x_2=\pm\frac{\pi}{3}+2k\pi~~ (k\in\mathbb{Z})\)(a) \(\sin x+\sqrt{3}\cos x=0\)
(b) \(\sin^2x+\sin2x-3\cos^2x=0\)
(c) \(6\sin^2x-\sin2x-2\cos^2x=1\)
Solutions: (a) \(x_1=-\frac{\pi}{3}+k\pi~~ (k\in\mathbb{Z})\); (b) \(x_1=\frac{\pi}{4}+k\pi,~ x_2=-\arctan3+k\pi~~ (k\in\mathbb{Z})\); (c) \(x_1=\frac{\pi}{4}+k\pi,~ x_2=-\arctan\frac{3}{5}+k\pi~~ (k\in\mathbb{Z})\)