高一正余弦函數(shù)試題及答案

一、單項(xiàng)選擇題(每題2分,共10題)1.函數(shù)\(y=\sinx\)的最小正周期是(）A.\(4\pi\)B.\(2\pi\)C.\(\pi\)D.\(\frac{\pi}{2}\)2.已知\(\sin\alpha=\frac{1}{2}\),且\(\alpha\)是第一象限角,則\(\cos\alpha\)的值為()A.\(-\frac{\sqrt{3}}{2}\)B.\(\frac{\sqrt{3}}{2}\)C.\(\pm\frac{\sqrt{3}}{2}\)D.\(\frac{1}{2}\)3.函數(shù)\(y=\cos(x+\frac{\pi}{2})\)的圖象與\(y=\sinx\)的圖象()A.相同B.關(guān)于\(x\)軸對(duì)稱C.關(guān)于\(y\)軸對(duì)稱D.關(guān)于原點(diǎn)對(duì)稱4.若\(\sin\theta+\cos\theta=\frac{1}{5}\),\(\theta\in(0,\pi)\),則\(\sin\theta-\cos\theta\)的值為()A.\(\frac{7}{5}\)B.\(-\frac{7}{5}\)C.\(\frac{49}{25}\)D.\(\frac{12}{25}\)5.函數(shù)\(y=\sin(2x+\frac{\pi}{3})\)的對(duì)稱軸方程是()A.\(x=\frac{k\pi}{2}+\frac{\pi}{12},k\inZ\)B.\(x=\frac{k\pi}{2}-\frac{\pi}{12},k\inZ\)C.\(x=k\pi+\frac{\pi}{12},k\inZ\)D.\(x=k\pi-\frac{\pi}{12},k\inZ\)6.已知\(\cos\alpha=-\frac{4}{5}\),\(\alpha\in(\frac{\pi}{2},\pi)\),則\(\sin2\alpha\)的值為()A.\(\frac{24}{25}\)B.\(-\frac{24}{25}\)C.\(\frac{12}{25}\)D.\(-\frac{12}{25}\)7.函數(shù)\(y=A\sin(\omegax+\varphi)\)(\(A\gt0,\omega\gt0\))的圖象上一個(gè)最高點(diǎn)為\((2,\sqrt{2})\),由這個(gè)最高點(diǎn)到相鄰最低點(diǎn)的圖象與\(x\)軸交于點(diǎn)\((6,0)\),則此函數(shù)的解析式是()A.\(y=\sqrt{2}\sin(\frac{\pi}{8}x+\frac{\pi}{4})\)B.\(y=\sqrt{2}\sin(\frac{\pi}{8}x-\frac{\pi}{4})\)C.\(y=\sqrt{2}\sin(\frac{\pi}{4}x+\frac{\pi}{4})\)D.\(y=\sqrt{2}\sin(\frac{\pi}{4}x-\frac{\pi}{4})\)8.函數(shù)\(y=\cos^{2}x-\sin^{2}x\)的最小正周期是()A.\(4\pi\)B.\(2\pi\)C.\(\pi\)D.\(\frac{\pi}{2}\)9.已知\(\sin(\alpha+\frac{\pi}{6})=\frac{1}{3}\),則\(\cos(\frac{\pi}{3}-\alpha)\)的值為()A.\(\frac{1}{3}\)B.\(-\frac{1}{3}\)C.\(\frac{2\sqrt{2}}{3}\)D.\(-\frac{2\sqrt{2}}{3}\)10.函數(shù)\(y=\sinx\)在區(qū)間\([-\frac{\pi}{6},\frac{5\pi}{6}]\)上的值域是()A.\([-\frac{1}{2},1]\)B.\([-\frac{1}{2},\frac{1}{2}]\)C.\([\frac{1}{2},1]\)D.\([-1,1]\)二、多項(xiàng)選擇題(每題2分,共10題)1.下列函數(shù)中,最小正周期為\(\pi\)的有（）A.\(y=\sin(2x+\frac{\pi}{3})\)B.\(y=\cos(2x-\frac{\pi}{6})\)C.\(y=\tan(2x+\frac{\pi}{4})\)D.\(y=|\sinx|\)2.對(duì)于函數(shù)\(y=\sinx\),以下說法正確的是()A.最大值為1B.圖象關(guān)于原點(diǎn)對(duì)稱C.最小正周期是\(2\pi\)D.在\([\frac{\pi}{2},\frac{3\pi}{2}]\)上單調(diào)遞減3.已知\(\sin\alpha=\frac{3}{5}\),\(\alpha\in(\frac{\pi}{2},\pi)\),則下列結(jié)論正確的是()A.\(\cos\alpha=-\frac{4}{5}\)B.\(\tan\alpha=-\frac{3}{4}\)C.\(\sin2\alpha=-\frac{24}{25}\)D.\(\cos2\alpha=\frac{7}{25}\)4.函數(shù)\(y=A\sin(\omegax+\varphi)\)(\(A\gt0,\omega\gt0\))的圖象的一個(gè)對(duì)稱中心為\((\frac{\pi}{3},0)\),則\(\varphi\)可能的值為()A.\(\frac{\pi}{6}\)B.\(-\frac{\pi}{6}\)C.\(\frac{2\pi}{3}\)D.\(-\frac{2\pi}{3}\)5.以下哪些是\(y=\cosx\)的性質(zhì)()A.偶函數(shù)B.圖象關(guān)于直線\(x=\pi\)對(duì)稱C.最大值為1D.在\([0,\pi]\)上單調(diào)遞減6.若\(\sin\alpha+\cos\alpha=\frac{1}{3}\),則()A.\(\sin\alpha\cos\alpha=-\frac{4}{9}\)B.\(\sin2\alpha=-\frac{8}{9}\)C.\((\sin\alpha-\cos\alpha)^2=\frac{17}{9}\)D.\(\sin\alpha-\cos\alpha=\pm\frac{\sqrt{17}}{3}\)7.函數(shù)\(y=\sin(2x-\frac{\pi}{6})\)的單調(diào)遞增區(qū)間是()A.\([k\pi-\frac{\pi}{6},k\pi+\frac{\pi}{3}],k\inZ\)B.\([k\pi+\frac{\pi}{3},k\pi+\frac{5\pi}{6}],k\inZ\)C.\([-\frac{\pi}{6},\frac{\pi}{3}]\)D.\([\frac{\pi}{3},\frac{5\pi}{6}]\)8.已知函數(shù)\(y=\cos(x+\varphi)\)的圖象關(guān)于原點(diǎn)對(duì)稱,則\(\varphi\)的值可能為()A.\(\frac{\pi}{2}\)B.\(-\frac{\pi}{2}\)C.\(\pi\)D.\(0\)9.下列等式成立的是()A.\(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\)B.\(\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\)C.\(\sin2\alpha=2\sin\alpha\cos\alpha\)D.\(\cos2\alpha=\cos^{2}\alpha-\sin^{2}\alpha\)10.函數(shù)\(y=\sinx\)與\(y=\cosx\)的圖象的交點(diǎn)坐標(biāo)為()A.\((\frac{\pi}{4},\frac{\sqrt{2}}{2})\)B.\((\frac{5\pi}{4},-\frac{\sqrt{2}}{2})\)C.\((\frac{\pi}{4},-\frac{\sqrt{2}}{2})\)D.\((\frac{5\pi}{4},\frac{\sqrt{2}}{2})\)三、判斷題(每題2分,共10題)1.函數(shù)\(y=\sinx\)在\([0,2\pi]\)上有兩個(gè)零點(diǎn)。()2.函數(shù)\(y=\cosx\)的圖象關(guān)于\(y\)軸對(duì)稱。()3.若\(\sin\alpha=\sin\beta\),則\(\alpha=\beta+2k\pi,k\inZ\)。（)4.函數(shù)\(y=A\sin(\omegax+\varphi)\)的振幅是\(A\)。()5.\(\sin(\frac{\pi}{2}+\alpha)=\cos\alpha\)。()6.函數(shù)\(y=\tanx\)的最小正周期是\(\pi\)。()7.函數(shù)\(y=\sin^{2}x\)的最小正周期是\(\pi\)。()8.若\(\cos\alpha=0\),則\(\alpha=\frac{\pi}{2}+k\pi,k\inZ\)。（)9.函數(shù)\(y=\sinx\)在\([\frac{\pi}{2},\pi]\)上的最大值是1。()10.函數(shù)\(y=\cos(x-\frac{\pi}{3})\)的圖象是由\(y=\cosx\)的圖象向右平移\(\frac{\pi}{3}\)個(gè)單位得到的。(）四、簡(jiǎn)答題(每題5分,共4題)1.求函數(shù)\(y=\sin(2x-\frac{\pi}{3})\)的單調(diào)遞減區(qū)間。答案:令\(2k\pi+\frac{\pi}{2}\leq2x-\frac{\pi}{3}\leq2k\pi+\frac{3\pi}{2},k\inZ\),解得\(k\pi+\frac{5\pi}{12}\leqx\leqk\pi+\frac{11\pi}{12},k\inZ\),所以單調(diào)遞減區(qū)間是\([k\pi+\frac{5\pi}{12},k\pi+\frac{11\pi}{12}],k\inZ\)。2.已知\(\sin\alpha=\frac{1}{3}\),\(\alpha\in(\frac{\pi}{2},\pi)\),求\(\cos\alpha\)和\(\tan\alpha\)的值。答案:因?yàn)閈(\sin^{2}\alpha+\cos^{2}\alpha=1\),\(\sin\alpha=\frac{1}{3}\),\(\alpha\in(\frac{\pi}{2},\pi)\),所以\(\cos\alpha=-\sqrt{1-\sin^{2}\alpha}=-\sqrt{1-(\frac{1}{3})^{2}}=-\frac{2\sqrt{2}}{3}\),\(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=\frac{\frac{1}{3}}{-\frac{2\sqrt{2}}{3}}=-\frac{\sqrt{2}}{4}\)。3.簡(jiǎn)述函數(shù)\(y=A\sin(\omegax+\varphi)\)(\(A\gt0,\omega\gt0\))中\(zhòng)(A\)、\(\omega\)、\(\varphi\)的意義。答案:\(A\)叫振幅,決定函數(shù)的最值;\(\omega\)決定周期\(T=\frac{2\pi}{\omega}\);\(\varphi\)叫初相,\(\omegax+\varphi\)叫相位,\(\varphi\)影響函數(shù)圖象左右平移。4.已知\(\sin\alpha+\cos\alpha=\frac{1}{5}\),求\(\sin2\alpha\)的值。答案:將\(\sin\alpha+\cos\alpha=\frac{1}{5}\)兩邊平方得\((\sin\alpha+\cos\alpha)^2=(\frac{1}{5})^2\),即\(\sin^{2}\alpha+2\sin\alpha\cos\alpha+\cos^{2}\alpha=\frac{1}{25}\),因?yàn)閈(\sin^{2}\alpha+\cos^{2}\alpha=1\),\(\sin2\alpha=2\sin\alpha\cos\alpha\),